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## Numbers and Algebra

### Basic Number Theory - Prime Numbers and Factorization

#### Q.02

'Prove using the binomial theorem that the following equation holds true: { }_{n} C_{0}+{ }_{n} C_{1}+{ }_{n} C_{2}+⋯+{ }_{n} C_{n}=2^n'

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'Common divisor (polynomials): A polynomial that divides evenly into all given polynomials, of two or more polynomials.'

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'(1) Prove that if m is a prime number, then d_{m}=m.\n(2) Prove by mathematical induction that for all natural numbers k, k^m-k is divisible by d_{m}.'

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'Prove that for all natural numbers n, the expression 4^{2n+1} + 3^{n+2} is a multiple of 13.'

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'Mathematics II\n(1) From (α-2)(α+3)=0, we get α=2,-3\n(2) From α=2, k=8 and from α=-3, k=-27\nTherefore k=8,-27'

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'Practice (1) Prove that when n is a natural number, 4^(2n+1) + 3^(n+2) is a multiple of 13.'

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'(1) Let $k ^ 3$ divided by $n$ have a quotient of $q$ and a remainder of $r$, then $k ^ 3 = qn + r (0≤r≤n-1)$ When $n$ and $k$ are coprime, $n$ and $k ^ 3$ are also coprime. Therefore, $r≠0$, so $1≤r≤n-1$ Dividing both sides of (1) by $n$ gives $\\frac{k ^ 3}{n} = q + \\frac{r}{n}$ From $1≤r≤n-1$, we know that $\\frac{1}{n}≤\\frac{r}{n}≤1-\\frac{1}{n}$. Therefore, 0 < $\\frac{r}{n}$ < 1, hence $\\left[\\frac{k ^ 3}{n}\\right] = \\left[q + \\frac{r}{n}\\right] = q$'

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'(2) Prove that for all k, k^m - k is divisible by d_m. [1] When k=1, 1^m - 1 = 0 and d_m ≠ 0, so 0 is divisible by d_m. Therefore, (1) holds. [2] Assuming that (1) holds for k=l, that is, l^m - l is divisible by d_m. Considering k=l+1, (l+1)^m - (l+1) ={m C_0 l^m + m C_1 l^(m-1) + m C_2 l^(m-2) + ... + m C_m - (l+1)} = {l^m - l} + {m C_1 l^(m-1) + m C_2 l^(m-2) + ... + m C_m-1 l} From the assumption, l^m - l is divisible by d_m. Also, d_m is the greatest common divisor of {m C_1, m C_2, ..., m C_(m-1)}, so these terms are also divisible by d_m. Therefore, (l+1)^m - (l+1) is divisible by d_m. Hence, when k=l+1, (1) also holds. From [1], [2], it can be concluded that (1) holds for all natural numbers k.'

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'Let p be a prime number and r be a positive integer, prove the following:\n(1) If x₁, x₂, ..., xᵣ are positive integers, then \\( \\left(x_{1}+x_{2}+\\cdots+x_{r}\\right)^{p}-\\left(x_{1}{ }^{p}+x_{2}^{p}+\\cdots+x_{r}^{p}\\right) \\) is divisible by p.\n(2) If r is not divisible by p, then \ r^{p-1}-1 \ is divisible by p.\n[Similar to Osaka University]'

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'Prove the following when p is a prime number:\n(1) For natural numbers k that satisfy 1 ≤ k ≤ p-1, p_kC_k is a multiple of p.\n(2) 2^p-2 is a multiple of force.\n[Tohoku Gakuin University]'

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"Let's define a word of length n as three letters (a, b, c) arranged horizontally n times. Here, n=1,2,3, … etc. For example, abbaca and caab are both different words of length 4. Among such words of length n, let's denote the ones containing an odd number of a's as xn, and the rest as yn. Find the values of xn and yn."

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'In exercise 55 (1) [1], when m=2, d_2 is the largest natural number that divides the binomial coefficient {2 C_1} = 2, so d_2=2, and d_m=m holds true. [2] When m is a prime number greater than or equal to 3, {m C_1} = m, therefore it is sufficient to show that {m C_2, m C_3, ..., m C_m - 1} are multiples of m. For k=2,3,…,m-1, {m C_k} = (m!) / (k!(m-k)!) = (m/k) * ((m-1)! / (k-1)!(m-k)!) = (m/k) * {m-1 C_k-1} thus, k * {m C_k} = m * {m-1 C_k-1}. Since m is a prime number greater than or equal to 3, and 2 ≤ k ≤ m-1, k and m are coprime. Hence, {m C_k} is a multiple of m. Therefore, d_m=m holds true. From [1], [2], if m is a prime number, then d_m=m.'

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'(3) \ m, n \ are natural numbers, and \ p \ is a prime number, so \ m, n, p \ are non-zero real numbers. Therefore, from (1), we have \ \\frac{1}{m} + \\frac{1}{n} = \\frac{1}{p} \. Also, in the equation \ a^{m} = b^{n} \, where \ 1 < a < b \, we have\ a^{m} = b^{n} > a^{n} \\text { which implies } a^{m} > a^{n} \\\\\\\The base \ a \ is greater than 1, so \ m > n \. Thus, from (2), we get \ m = p^{2} + p, n = p + 1 \, and therefore\\[ a^{p^{2} + p} = b^{p + 1} = (a b)^{p} \\]\\\\'

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'Let the solutions to (1) be α and β, and let f(x)=x^2+2ax+a-1. The condition for α and β to be between the two solutions of (2) is that, under the conditions of (3), f(α)<0 and f(β)<0.'

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'When non-zero real numbers $x, y, z$ satisfy $3^{x}=2^{y}=5^{z}=(\x0crac{6}{5})^{7}$, find the value of $\x0crac{1}{x}+\x0crac{1}{y}-\x0crac{1}{z}$.'

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'Assume integers a and b are not multiples of 3, and let f(x) = 2x^3 + a^2x^2 + 2b^2x + 1. Prove that there does not exist an integer x that satisfies f(x) = 0.'

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'Given $4^{x}=6^{y}=24$, find the value of $\\frac{1}{x}+\\frac{1}{y}$.'

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"Let's reflect on the concept of repeated roots! In mathematics, repeated roots refer to the case when b^2-4ac=0 in the quadratic equation ax^2+bx+c=0. In the formula for finding the roots of a quadratic equation, x=-b±√(b^2-4ac)/(2a), when b^2-4ac=0, both √(b^2-4ac) and -√(b^2-4ac) are 0, resulting in the root x=-b/(2a)."

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'Let n be a natural number greater than or equal to 3, prove the inequality 4^{n}>8 n+1 (A).'

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'Which of the following is a factor of the polynomial 2x^3+5x^2-23x+10?'

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'The necessary and sufficient condition for the existence of real numbers x, y satisfying the equations x² - xy + y² = k and x + y = 1 is k ≥ 0.'

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'Basic 57: Finding coefficients from divisible conditions'

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'Express the following sets of numbers in order using inequality symbols.'

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'If you deposit 1 million yen with an annual interest rate of 1% compounded annually, in how many years will the total amount of principal and interest first exceed 1.1 million yen? You may use common logarithm table.'

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'2020 Shibuya Education Institute Makuhari Middle School Arithmetic First Exam'

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'Arrange integers greater than 1 that are neither square numbers nor cube numbers in ascending order. 2,3,5,6,7,10,11, \\cdots \\cdots What is the 2020th integer when counted from the smallest?'

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'(4) When the side length of the black square is between 1 cm and 100 cm, the number of white squares is at least 8 ((1+1) x 4 = 8) and at most 404 ((100+1) x 4 = 404). A number that cannot be expressed as the sum of consecutive integers, besides 1, is an integer that does not have odd divisors. This type of number can be expressed as a product of primes, such as 2 x ・・・ x 2. Therefore, within the mentioned range, there are 8 (pieces), 16 (pieces), 32 (pieces), 64 (pieces), 128 (pieces), and 256 (pieces), with the corresponding side lengths of the black squares being 1 cm, 3 cm, 7 cm, 15 cm, 31 cm, and 63 cm, respectively.'

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'The ambassador is well received every time he visited the prime minister.'

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'Regarding question 5, the part underlined as d, the exchange rate between foreign currency and another currency is referred to as foreign exchange rate. For the following statements X and Y concerning this matter, select the correct combination from the options below to answer.'

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'(2) When 3240 is expressed as a product of prime numbers, 3240 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 5.'

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'Arrange the integers greater than 1 that are not perfect squares in ascending order such as 2, 3, 5, 6, 7, 8, 10, ..., What is the 300th integer when counting from the smallest?'

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'Explain the Fibonacci sequence and determine what value the ratio of consecutive terms in this sequence converges to.'

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'(1) Prove the inequality $2^{n}>\x0crac{1}{6} n^{3}$ holds using the binomial theorem. (2) Find the value of $\\lim _{n \\rightarrow \\infty} \\frac{n^{2}}{2^{n}}$.'

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'Prove that for two integers a and b, if a+b and ab are coprime, then a and b are coprime.'

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'Find the number of natural numbers less than or equal to 56 that are coprime to 56.'

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'Let a and b be natural numbers, where a + b = p + 4 and ab^{2} = q. Find prime numbers p and q that satisfy these conditions.'

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'Prove that among any 26 distinct integers chosen from 1 to 50, there must be a pair of numbers whose sum is 51.'

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'Prove that n^{2}+1 is a multiple of 5 if and only if the remainder of n divided by 5 is 2 or 3.'

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'A prime number is a natural number greater than 1, having no positive divisors other than 1 and itself, whereas a number that is not prime is called a composite number. For example, 2, 3, 5, 7, 11, etc. are prime numbers, while 4, 6, 8, 9, etc. are composite numbers.'

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'Under the given conditions, when p=3k+2, the natural number p that makes p, 2p+1, and 4p+1 all prime numbers is p=3. For prime numbers p greater than or equal to 5, it is evident that either 2p+1 or 4p+1 will be a multiple of 3.'

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'Determine the truth or falsehood of the following propositions:\n(2) The positive divisors of 28 are 1, 2, 4, 7, 14, and 28, which is a total of 6 divisors. Therefore, this is a true proposition.\n(3) When n=36, n is a multiple of 4 and 6, but not a multiple of 24. Therefore, this is a false proposition (with n=36 as a counterexample).'

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'Prove that the product of consecutive integers is a multiple of 2.'

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'I\'m confused about how to approach prime number problems. The definition of prime numbers, "Integers greater than 2 that have no positive divisors other than 1 and themselves," is simple. The key is how to use this definition effectively. First, let\'s understand the following properties (1) and (2): (1) The divisors of a prime number p are ±1 and ±p (there are 2 positive divisors: 1 and p), (2) Prime numbers are greater than 2, and the only even prime number is 2. Additionally, all prime numbers greater than 3 are odd. By using the property that "the divisors of prime number p are ±1 and ±p", we can consider four cases (A) to (D) for when (n-3)(n-9) is a prime number p. Pay attention to the relationship n-9<n-3 and 1<p,-p<-1, where only (B) n-9=1 and (C) n-3=-1 are possible. In particular, be careful of mistakes like n-9=-1 in negative cases.'

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'Answer the following questions regarding the number of elements in a set and the number of cases:\n(1) Find the number of positive divisors of the following number. 360 = 2^3 * 3^2 * 5\n(2) Find the number of terms in the expansion of the following polynomial. (a+b)(p+q+r)(x+y)\n'

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'For all natural numbers n, n^2-2n-3 ≠ 0, false\nFor some real numbers x, y, x^2 + 2xy + y^2 ≤ 0, true'

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'Find the smallest positive integer whose number of positive divisors is 28.'

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'Find the greatest common divisor and least common multiple of 72 and 120.\nDivide by common prime factors among the 12 numbers.\nFor example, continue dividing by 2.\n2) 72 \t 120\n2) 36 \t 60\n2) 18 \t 30\nCalculate the greatest common divisor and least common multiple.'

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'Find all natural numbers p such that the three numbers p, 2p+1, 4p+1 are all prime.'

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'Let p and q be prime numbers with p<q. Also, let m and n be positive integers such that m≥3 and n≥2. Assume that among the integers from 1 to p^m * q^n, the number of integers that are multiples of either p or q is 240. Find the set of (p, q, m, n) that satisfy these conditions.'

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'Basic Example 106 Number of Positive Divisors\n(1) Find the number of positive divisors of 630.\n(2) If a natural number N is factorized into prime factors, where the prime factors include p and 7, and there are no other prime factors. Furthermore, N has 6 positive divisors, and the sum of positive divisors is 104. Find the values of the prime factor p and the natural number N.'

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'(2) Let a be a positive integer, and let p = a^2 + 1 be a prime number. Then n^2 + 1 is a multiple of p if and only if the remainder when n is divided by p is a or p - a.'

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'Prove that for two coprime integers a and b, a+b and ab are also coprime.'

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'Find the greatest common divisor and least common multiple of 2 integers or 3 integers using prime factorization.\n(1) 168, 378\n(2) 65,156,234'

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'Find the number of natural numbers less than 432 that are coprime to 432.'

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'(2) Prove that if a natural number P is not divisible by 2 or 3, then P^2-1 is divisible by 24.'

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'Find the number of natural numbers less than 735 that are coprime to 735.'

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'To find the remainder when 13 to the power of 15 is divided by 5.'

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'Let a and b be natural numbers. Prove that if ab is a multiple of 3, then either a or b is a multiple of 3.'

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'When N = 250! is factorized, answer the following questions: (1) Find the number of prime factors 5. (2) When calculating N, how many consecutive zeros will appear at the end?'

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'Paint in the order of D → A → B → C → E. There are 6 ways to paint D → A → B (3!). For each of these, there is 1 way to paint C, and 1 way to paint E. Therefore, the total number of painting ways is 6 × 1 × 1 = 6.'

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'Find the number of positive divisors of the positive integer 756.'

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'Problem (2) decomposes the natural number N into prime factors, where the prime factors are p and 5, and there are no other prime factors. In addition, N has 8 positive divisors, and the sum of positive divisors is 90. Find the values of the prime factor p and the natural number N.'

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'When the prime factorization of a natural number N is N=p^a * q^b * r^c ......, the number of positive divisors of N is (a+1)(b+1)(c+1) ......'

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'There is a quiz to guess the age: My age leaves a remainder of 1 when divided by 3, a remainder of 4 when divided by 5, and a remainder of 1 when divided by 7. Please guess my age. It is less than 105 years old.'

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'Prove that the product of four consecutive integers n(n+1)(n+2)(n+3) is a multiple of 24.'

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'Prove that 2n-1 and 2n+1 are coprime for any natural number n.'

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'Prove that when a natural number P is not divisible by either 2 or 3, then P^2-1 is divisible by 24.'

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'Prove the condition for all three numbers to be prime'

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'Let n be a natural number. Find all values of n that make the following expressions prime:\n(A) n^2 - 2n - 24\n(B) n^2 - 16n + 28'

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'What are the challenges in discovering large prime numbers?'

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'Find the smallest natural number n such that √(378n) becomes a natural number.'

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'When an integer can be expressed as the product of several integers, each integer in the product is called a factor of the original integer. Factors that are prime numbers are called prime factors, and expressing a natural number in the form of a product containing only prime numbers is called prime factorization.'

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'Find the smallest positive integer for which the number of positive divisors is 28.'

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'Finding prime numbers (Sieve of Eratosthenes)\nIf a natural number n is not divisible by all prime numbers less than or equal to its square root, then n is a prime number.\nUsing this rule, consider a method to find all prime numbers less than or equal to 50.'

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'Fundamentals 11: Factoring by extracting common factors'

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'Prove that the product of two consecutive integers is a multiple of 2.'

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'When two integers a and b have no common prime factors, their greatest common divisor is 1. If the greatest common divisor of two integers a and b is 1, then a and b are said to be coprime.'

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"Let's summarize the basic steps of prime factorization. To apply the formulas for prime factorization:"

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'When rolling a die twice, how many ways are there for the product of the outcomes to be a multiple of 12?'

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'From a≥1 and b≥1, it follows that a+b>a+b-1≥1. Furthermore, because a+b-1 is a prime number, a+b-1=1. Therefore, a+b=p. Since a≥1 and b≥1, we have a=1 and b=1. Thus, from (2), we get p=2, which is a prime number. Hence, the values of a and b that make p a prime number are a=1 and b=1.'

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'Calculate how many times 60! can be divided by 3 and how many consecutive 0s will appear at the end of 50! when calculated?'

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'(1) Proof: Assume that the integer n is not a multiple of 3, then n can be represented as 3k±1 (k is an integer). So, n^2-1 = (3k±1)^2-1 = 9k^2±6k+1-1 = 9k^2±6k = 3(3k^2±2k) which must be a multiple of 3.'

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'Find the greatest common divisor of the following pairs of integers using the Euclidean algorithm: (1) 221, 91 (2) 418, 247 (3) 1501, 899'

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'Find all values of p for which 51, 2p+1, and 4p+1 are prime numbers. Check if 2p+1 and 4p+1 are prime when p is a prime number.'

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'Prove that for two natural numbers a and b, if a and b are coprime, then a+b and ab are also coprime.'

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'Prove that for any natural numbers a and k, a and ka+1 are coprime.'

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'Chapter 4 Divisors and Multiples -235 EX 50 Let n be a natural number greater than or equal to 2, then prove that n^4 + 4 is not a prime number. [Miyazaki University]'

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'What kind of problems are good to work on after solving basic examples and standard examples?'

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'Please answer the following questions: (1) Calculate the result of 60!, and determine the maximum number of times it can be divided by 3. (2) Calculate 50!, and determine how many consecutive zeros appear at the end.'

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'Prove the following for natural numbers a, b:\n(1) If a and b are coprime, then a^2 and b^2 are coprime.\n(2) If a+b and ab are coprime, then a and b are coprime.'

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'Prove that for any natural number n greater than or equal to 2, n^4+4 is not a prime number.'

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'Use the symbols $\\subset ,=$ to describe the relationship between the two sets $A, B$. A=\\{n \\mid n is a prime number less than or equal to 7 \\}, \\quad B=\\{2n-1 \\mid n=2,3,4\\}'

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'For the following mathematical questions: (1) Using the quotient of dividing 10 by 2, dividing 4 by 2, and dividing 2 by 2, with the method of counting the number of multiples of 2, what is the maximum number of times that 10! can be divided by 2? (2) Using the quotient of dividing 10 by 5, calculate 10! and determine how many consecutive zeros appear at the end?'

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'(1) Find the number of positive divisors of 720.\n\n(2) Decompose a natural number N into prime factors, where the prime factors are 2 and 3, with no other prime factors. Also, it is known that N has exactly 10 positive divisors. Find all such natural numbers N.'

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'A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a natural number greater than 2 that is not a prime number.'

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'How many strings can be formed using all 8 letters of TANABATA?'

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'Question A (2) Find two natural numbers denoted as 6m and 6n, where m and n are coprime natural numbers. Since 6m>6 and 6n>6, we have m>1 and n>1. Given 4536=6m·6n, we get mn=126. As mn is not a perfect square, m cannot equal n, thus 1<m<n. Solving for pairs of m and n that satisfy this condition, we get (m, n) = (2,63), (3,42), (6,21), (7,18), (9,14). Among these pairs, the coprime ones are (2,63), (7,18), (9,14). Therefore, the two required natural numbers are 12,378 or 42,108 or 54,84.'

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'How do you convert the binary number 101 to decimal?'

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"When the 8 letters of the word 'addition' are arranged horizontally in a single row, how many possible ways are there to arrange them?"

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'Master the method of determining multiples and conquer example 85!'

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'Find the smallest natural number that has 8 positive divisors.'

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'(1) Let n be a natural number. Find all values of n for which the following expressions result in a prime number. (a) n^{2}+6 n-27 (b) n^{2}-16 n+39 (2) Let a, b be natural numbers, and let p=a^{2}-a+2 a b+b^{2}-b. Find all values of a, b for which p is a prime number.'

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'(1) How many natural numbers N exist such that they have 3 digits when represented in base-5?'

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'Find the smallest natural number with 4 positive divisors.'

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'Answer: Math section 50 omitted 51 (1) {1,2,3,4,5,6,7,9,12,18} (2) {1,2,3,6}'

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'Find all values of p for which p, 2p+1, and 4p+1 are all prime numbers.'

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'Permutation with order determined. Standard 20 permutation with order determined.'

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'Prove that a and k a+1 are coprime when a and k are natural numbers.'

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'When we divide Example 83 into maximum and minimum values, we get the following results.'

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'Problem of finding integer solutions to a linear Diophantine equation (3) (using the Euclidean algorithm).'

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"Let's review how to find integer solutions for linear Diophantine equations! When integer solutions are not easily found, you can use the method of successive divisions. By retracing the calculations of the method of successive divisions in reverse, you can find integer solutions."

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'Assuming a and b are not relatively prime, i.e., a and b have a common prime factor p, then a=pk, b=pl (k, l are natural numbers).'

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'When rolling two dice at the same time, how many ways can the number 1 not appear on any of the dice?'

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'There is a point P on the x-axis. When a six-sided die is rolled and a multiple of 6 appears, P moves forward 1 unit in the positive direction of the x-axis, and when a non-multiple of 6 appears, P moves 2 units in the negative direction of the x-axis. When the die is rolled 4 times, the probability that the point P, starting from the origin, is at the point x=-2 is A, and the probability that it is at the origin is B.'

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'Find the number of positive divisors and their sum of 648.'

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'Find the largest three-digit natural number that leaves a remainder of 5 when divided by 14 and a remainder of 7 when divided by 9.'

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'Find the maximum value of n for EX children and the corresponding values of a, b'

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'Euclidean algorithm\nFor natural numbers a and b, if a is divided by b and the remainder is r, then the greatest common divisor of a and b is equal to the greatest common divisor of b and r.\nBy repeatedly using this method, we can find the greatest common divisor of two natural numbers. This method is called the Euclidean algorithm or simply the division algorithm.\nFor example, finding the greatest common divisor of 319 and 143\nBy observing the division of 319 by 143 resulting in the equation 319=143*2+33, according to the theorem, instead of finding the greatest common divisor of 319 and 143, we can find the greatest common divisor of the divisor 143 and the remainder 33. Continuing this operation, the remainders will decrease. Moreover, since the remainder is greater than or equal to 0, eventually the remainder will become 0. When the remainder becomes 0, the divisor at that step is the desired greatest common divisor.'

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'Mathematics A\nTR\n(1) Using congruence equations, find the following:\nFind the remainder when 12^{1000} is divided by 11\nFind the unit digit of 13^{81}\n(2) Prove using congruence equations that if integers a, b, c satisfy a^2+b^2=c^2, then at least one of a and b is a multiple of 3.'

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'Divide 5390 by a natural number n such that the remainder is 0 and the quotient is a square of a natural number. Find the minimum value of n that satisfies this condition.'

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'Find the smallest natural number that has 8 positive divisors.'

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'1. Let n be an integer. Find all values of n such that (n-4)(n+8) is a prime number. 2. Let a and b be distinct natural numbers. Find prime numbers p and q that satisfy both equations ab=p and a+b=q.'

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'Factorize the natural number N, where the prime factors are 3 and 5, and there are no other prime factors. Moreover, N has exactly 6 positive divisors. Find all natural numbers N that satisfy these conditions.'

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'Explain the proof by contrapositive method, and prove the following proposition T using contrapositive:'

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'Let a and b be natural numbers. Prove the following: (1) If a and b are coprime, then a^{2} and b^{2} are coprime. (2) If a+b and ab are coprime, then a and b are coprime.'

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'Prove that for any natural number a, a and a+1 are coprime.'

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'Find the smallest 4-digit natural number that leaves a remainder of 8 when divided by 23 and a remainder of 5 when divided by 15.'

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'Multiply 150 by a two-digit natural number n in order to make it a square of a certain natural number. Find the maximum value of n that satisfies this condition.'

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'When rolling three dice at the same time, how many ways are there for all three dice to show odd numbers?'

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'(1) \\\\\\ (72^{\\circ} \\\\\\\\\n(2) \\\\\\\n(\\frac{\\sqrt{5}-1}{2} \\\\\\\\\n(3) \\\\\\\n(\\frac{\\sqrt{5}+1}{4}'

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'Find the number of elements in the following sets within the natural numbers less than 500:\n(1) Set of numbers divisible by 3\n(2) Set of numbers divisible by 3, 5, and 7\n(3) Set of numbers divisible by 3 but not by 5\n(4) Set of numbers not divisible by either 3 or 5\n(5) Set of numbers divisible by 3 but not by 5 or 7'

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'(1) Find the number of positive divisors of 1800.\n\n(2) When a natural number N is prime factorized, its prime factors are 3 and 5, with no other prime factors. Also, N has exactly 6 positive divisors. Find all such natural numbers N.'

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'When non-zero real numbers x, y, z satisfy 2^{x}=5^{y}=10^{\x0crac{z}{2}}, find the value of \x0crac{1}{x}+\x0crac{1}{y}-\x0crac{2}{z}.'

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'Determine the number of distinct real solutions of the equation x^3-3x^2-9x+k=0.'

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'Let \ \\omega \ be one of the imaginary solutions of the equation \ x^{3}=1 \. Then, \ \\frac{1}{\\omega}+\\frac{1}{\\omega^{2}}+1=\\square, \\omega^{100}+\\omega^{50}=\\square \.'

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'Find the general term of the sequence 1, 17, 35, 57, 87, 133, 211, ...'

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'If non-zero real numbers x, y, z satisfy 2^{x}=5^{y}=10^{\x0crac{z}{2}}, find the value of \x0crac{1}{x}+\x0crac{1}{y}-\x0crac{2}{z}.'

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'Find the value of p when the sum of irreducible fractions with prime numbers as denominators between 1 and 10 is 198.'

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'Using the binomial theorem, find the following values:'

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'3. \ { }_{n} \\mathrm{C}_{0}+{ }_{n} \\mathrm{C}_{1}+{ }_{n} \\mathrm{C}_{2}+\\cdots \\cdots+{ }_{n} \\mathrm{C}_{n}=2^{n} \'

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'Consider the sequence $a_{1}=1, a_{2}=1, a_{n+2}=a_{n+1}+a_{n}$, find the general term of this sequence.'

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'Choose one of the following 0-5: \n(0) p_{4}<p_{5}\n(1) p_{4}=p_{5}\n(2) p_{4}>p_{5}'

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'There exist exactly two complex numbers z=x+yi (where x, y are real numbers) such that the square of z is equal to 8i. Find these z.'

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'Assuming it is a geometric progression, the common ratio is \\frac{6}{3}=2. If the nth term is 1500, then 3* 2^{n-1}=1500. As a result, 2^{n-1}=500, 500=2^{2}* 5^{3}, hence there is no natural number n that satisfies this equation. Therefore, it cannot be a geometric progression.'

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'Prove that for all positive integers n, 3^(3n-2)+5^(3n-1) is a multiple of 7.'

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'Prove that for all positive integers n, 3^{3n-2}+5^{3n-1} is a multiple of 7.'

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'Let k be a positive integer. Find all values of k such that there is exactly one integer n satisfying 5n^{2}-2kn+1<0.'

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'For the two equations $x^{2}-x+a=0, x^{2}+2ax-3a+4=0$, determine the range of values for the constant $a$ so that the following conditions are met:\n(1) Both equations have real solutions\n(2) At least one of them does not have real solutions\n(3) Only one of them has real solutions'

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'Express the symbols and ways of representation of set 44.'

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'Find the maximum and minimum values of 2x+y when real numbers x and y satisfy x²+y²=2. Also, determine the values of x and y at that time.'

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'Find the range of values for the constant k so that the quadratic equation x² + (2k-1)x + (k-1)(k+3) = 0 has real roots.'

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'Among three consecutive natural numbers, the square of the smallest number is equal to the sum of the other two numbers. Find these three numbers.'

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'(1) The meaning of "big" is not clear, so it is not possible to determine if it is true or false. Therefore, it is not a proposition.'

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'Find the range of values for the constant $a$ such that the quadratic equation $x^{2}+(a-3)x-a+6=0$ has no real solutions.'

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'Find the number of intersection points between the parabola y = 2x^2 + 3x - a + 1 and the x-axis using the constant a.'

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'Find the solutions to the factored quadratic inequalities. Find the solutions to the following inequalities.'

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'Please prove that the expression $(a-b)(b-c)(c-a)$ has factors.'

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'Find a condition for having one solution greater than p and one solution less than p.'

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'(4) Let a₁, b₁ be coprime positive integers, and let a₂, b₂ also be coprime positive integers. Define sets Q₁ and Q₂ as\nQ₁={z | z is a complex number represented as (cos(2𝑎_{1}/𝑏_{1}π) + i sin(2𝑎_{1}/𝑏_{1}π))^k using an integer k}\nQ₂={z | z is a complex number represented as (cos(2𝑎_{2}/𝑏_{2}π) + i sin(2𝑎_{2}/𝑏_{2}π))^k using an integer k}\nand define set R as\nR={z | z is a complex number represented as a product of elements from set Q₁ and set Q₂}. If b₁ and b₂ are coprime, the number of elements n(R) in set R is square. If b₁ and b₂ are not coprime, and we denote their greatest common divisor as d, then the number of elements n(R) in set R is circle.'

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'Sum of infinite series using recurrence relation'

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'Prove \\((k+1)!\\)^{2} = \\((k+1) \\cdot k!\\)^{2} = (k+1)^{2} \\cdot (k!)^{2} \\geqq (k+1)^{2}(k+1)^{k-1} = (k+1)^{k+1} \\).'

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'(2) Let l and k be coprime natural numbers. Prove that the complex numbers z^l, z^2l, z^3l, ..., z^kl are all distinct.'

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'8 (1) Incorrect\n(2) Incorrect\n(3) Incorrect\n(4) Correct'

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'Here, the nth prime number Pn satisfies Pn>n, and for all natural numbers k satisfying 1 ≤ k ≤ n, k can be expressed using prime numbers p1, p2, ..., pn as k = p1^m1(k)×p2^m2(k)×...×pn^mn(k) [where m1(k), m2(k), ..., mn(k) are integers greater than or equal to 0 and less than or equal to n]. Therefore, 1/k = 1/(p1^m1(k)×p2^m2(k)×...×pn^mn(k))'

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'Deriving from the condition of C winning the competition, we get $\\frac{2 p^{2}}{p^{2}-p+2} \\geqq \\frac{1}{3}$. Since $p^{2}-p+2=\\left(p-\\frac{1}{2}\\right)^{2}+\\frac{7}{4}>0$, clearing the denominator and simplifying gives $5 p^{2}+p-2 \\geqq 0$. Solving this inequality yields $p \\leqq \\frac{-1-\\sqrt{41}}{10}, \\frac{-1+\\sqrt{41}}{10} \\leqq p$. Note that $\\frac{-1-\\sqrt{41}}{10}<0$, and since $6<\\sqrt{41}<7$ implies $\\frac{1}{2}<\\frac{-1+\\sqrt{41}}{10}<\\frac{3}{5}$. Hence, with $0<p<1$, we have $\\frac{-1+\\sqrt{41}}{10} \\leqq p<1$. Next, we find the natural number $n$ that satisfies the condition $\\frac{n-1}{100}<\\frac{-1+\\sqrt{41}}{10} \\leqq \\frac{n}{100} \\cdots$ (1). Solving yields $54<-10+\\sqrt{4100}<55$. Since $n$ is monotonically increasing, the smallest $n$ satisfying (1) is $55$. Therefore, the minimum value of the required $N$ is 55.'

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'In the case of polynomials, we can also use factorization to find the greatest common divisor and the least common multiple, similar to the case of integers.'

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'When plucked, a string at half length produces a sound one octave higher. Here, the ratio of string lengths between the C and the higher octave C is divided into 12 equal parts, forming the 12-tone equal temperament. This is a commonly used scale.'

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'Find the remainder when 29^51 is divided by 900.'

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'When positive real numbers x, y satisfy 9x^2 + 16y^2 = 144, the maximum value of xy is √.'

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'Translate the given text into multiple languages.'

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'Find the polynomial x such that when divided by x^2+1, the remainder is 3x+2, and when divided by x^2+x+1, the remainder is 2x+3, with the minimum degree of x being 48.'

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'Find all positive integers n such that n^{n}+1 is divisible by 3.'

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'Determine the values of constants a and b such that f(x)=a x^{n+1}+b x^{n}+1 is divisible by (x-1)^{2}, where n is a natural number.'

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'Let \ p \ be a prime number, and let an integer \ r \ satisfy \ 1 \\leqq r \\leqq p-1 \. Show that \ p_r \ is divisible by \ p \.'

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'If a, b are prime numbers and the quadratic equation 3 x^{2}-12 a x+a b=0 has two integer solutions, find the values of a, b and the integer solutions.'

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'(2) \ \\sqrt{d}=\\sqrt{a b^{2} c^{3}}=b c \\sqrt{a c} \ The condition for \ \\sqrt{d} \ to be an integer is that the product of \a c\ must be a perfect square. Among such natural numbers \\(a, c(a>c>1)\\), the smallest is given by \ a=2^{3}, c=2 \ Choosing \b=3\ gives \d=2^{3} \\cdot 3^{2} \\cdot 2^{3}=576\.'

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'If the remainder of dividing P(x) by (x-1)^{2} is a constant, find the remainder when dividing P(x) by (x-1)^{2}(x+1).'

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'In a square on the complex plane, if one pair of adjacent vertices are point 1 and point 3+3i, find the complex numbers representing the other two vertices.'

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'Translate the given text into multiple languages.'

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'Real and pure imaginary conditions of complex number z\nLet z=a+bi (a, b are real numbers)\n• z is real ⇔ z=̄z\nSince ̄z=z holds, a-bi=a+bi, which implies -b=b, so b=0, hence z=a, and z is real.\nConsidering this on the complex plane, point z and point ̄z are two symmetric points about the real axis, these two points coincide only on the real axis, therefore z is real.\n• z is pure imaginary ⇔ ̄z=-z and z≠0\nSince ̄z=-z and z≠0 holds, a-bi=-a-bi, implying a=-a, so a=0, therefore z=bi, and since z≠0, therefore b≠0, so z is pure imaginary.\nConsidering this on the complex plane, point ̄z and point -z are two symmetric points about the imaginary axis, these two points coincide only on the imaginary axis, except for the origin O, all other points are pure imaginary, so z is pure imaginary.'

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'Prove that for complex numbers $z$ and $w$ satisfying $|z|=|w|=1, zw \\neq 1$, the expression $\\frac{z-w}{1-zw}$ is a real number.'

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'There are how many 5-digit numbers that can be formed by arranging all five of the numbers 0, 1, 2, 3, 4? The 40th number when these integers are arranged in ascending order is , and 32104 is the number, when arranged in ascending order, in what position?'

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'Prove that for natural numbers n, k satisfying 2 ≤ k ≤ n-2, the binomial coefficient C(n, k) > n.'

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'Using the Sieve of Eratosthenes, prove that there are more than 750 non-prime integers below 1000.'

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"Please explain the content of Brahmagupta's axioms."

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'The maximum value of n is obtained by calculating the number of zeros at the end of 50!, which is equal to the number of prime factor 5 when 50! is prime factorized. Among the natural numbers from 1 to 50, the number of multiples of 5 is 10 (the number of multiples of 5^2 is 2, as 50 divided by 5^2 is 2). Since there are no multiples of 5^n (n ≥ 3), the number of prime factor 5 is 10+2=12. Hence, the maximum value of n to be found is 12.'

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'Prove that for any natural number n, f(n) = 5^{3n} + 5^{2n} + 5^n + 1. When n is not a multiple of 4, f(n) is a multiple of 13.'

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"Describe the steps of Euclid's algorithm and provide a specific example, please."

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'If a and b are coprime, and a k is a multiple of b, then k is also a multiple of b.'

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'For a prime number p, find the minimum value of p such that n = p^14 and n ≥ 1900.'

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'Prove that for any natural number n, n^5 - n is a multiple of 15.'

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'Exercise 6 III-> Book p .59 \\[ x = \\sqrt{12 + 2 \\sqrt{35}} = \\sqrt{(7 + 5) + 2 \\sqrt{7 \\cdot 5}} = \\sqrt{7} + \\sqrt{5} \\\\\\ y = \\sqrt{12 - 2 \\sqrt{35}} = \\sqrt{(7 + 5) - 2 \\sqrt{7 \\cdot 5}} = \\sqrt{7} - \\sqrt{5} \\\\\\ \\sqrt{\\frac{x}{y}} = \\sqrt{\\frac{\\sqrt{7} + \\sqrt{5}}{\\sqrt{7} - \\sqrt{5}}} = \\sqrt{\\frac{(\\sqrt{7} + \\sqrt{5})^{2}}{(\\sqrt{7} - \\sqrt{5})(\\sqrt{7} + \\sqrt{5})}} = \\sqrt{\\frac{(\\sqrt{7} + \\sqrt{5})^{2}}{7 - 5}} = \\sqrt{\\frac{(\\sqrt{7} + \\sqrt{5})^{2}}{2}} = \\frac{\\sqrt{7} + \\sqrt{5}}{\\sqrt{2}} = \\frac{(\\sqrt{7} + \\sqrt{5}) \\sqrt{2}}{(\\sqrt{2})^{2}} = \\frac{\\sqrt{14} + \\sqrt{10}}{2} \\]'

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'Natural numbers greater than 2 can be factorized into prime factors.'

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'Let p be a prime number. Find all pairs of natural numbers (n, k) satisfying k ≤ n and such that the binomial coefficient C(n, k) = p.'

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'Find all the prime number triples (a, b, c) where 40-a-8 and b-c-8 are prime.'

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'Numbers that are greater than 125 and multiples of 5 include 150, 155, 160, 165, 130, etc. When factorizing 165!, what is the number of prime factor 5?'

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'How many natural numbers from 1 to 100 are divisible by 2, 3, and 5? How many natural numbers are divisible by 2, 3, or 5? How many numbers are divisible by 2 but not by 3 or 5?'

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'(2) Prove that there exist non-prime numbers among a, b, c.'

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'Please solve the problem about Gaussian notation and quadratic inequalities.'

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'Find the values of the natural number n for which both n and n^{2}+2 are prime numbers.'

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"Provide examples of composite numbers for which the converse of Fermat's Little Theorem 'If the coprime integer a does not satisfy a^{p-1} ≡ 1 (mod p), then p is not a prime (but a composite)' holds true: 9, 35."

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'Translate the given text into multiple languages.'

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'Find the divisors of the following numbers. (1) 36 (2) 14 (3) Is 12345 a multiple of 3 or 9? (4) Are 91 and 144 relatively prime?'

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'Find all odd numbers a greater than 423 and less than 9999 for which (a^2 - a) is divisible by 10000.'

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'Prove that composite numbers always have prime numbers as factors.'

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'Translate the given text into multiple languages.'

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'In (3) (2), if we remove the distinction between A, B, and C, then the same things can be combined in 3! ways each, so 1680 ÷ 3! = 1680 ÷ 6 = 280 (ways)'

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'A prime number p satisfies the condition: m² - n² = p. Prove that there exists a unique pair of natural numbers (m, n) that satisfy this condition.'

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'Assume that p is a prime number larger than 3, and p + 4 is also a prime number.'

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'Prove the formula to find the nth Catalan number (Catalan number Cn). Also, find the Catalan number when n=4.'

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'Prove the following proposition: If an integer n is not a multiple of 3, then n² is also not a multiple of 3.'

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'Challenge accepted. The solution also contains =.'

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'Example 49 | Classification of Integers by Remainder\nProve the following:\n(1) For any integer n, n^{4}+5 n^{2} is a multiple of 3.\n(2) The remainder is never 3 when squaring an integer and dividing by 5.'

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'Find the number of elements in the following sets among natural numbers less than 500.'

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'Find all the factors of the given numbers 25 and 36.'

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'(1) Calculate the result of 20 factorial, how many times can it be divided by 2.\n(2) Calculate 25 factorial, how many consecutive zeros will appear at the end.'

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'Let $n=2^{m-1}\\left(2^{m}-1\\right)(m=2,3,4, \\cdots \\cdots)$. Prove that $T(n)=n$ when $2^{m}-1$ is a prime, and use $1+2+\\cdots \\cdots+2^{m-1}=2^{m}-1$.'

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'If ab is a multiple of prime number p, then either a or b is a multiple of p.'

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'Divisors and Multiples Problem: Find the number of positive divisors of a natural number N. When the prime factorization of a natural number N is N=p^a q^b r^c ... ..., the number of positive divisors of N is'

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'Prove the conditions for the existence of integer solutions to the 99 1 1 indeterminate equation'

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'Since (3k + 1)(3k + 2) is the product of two consecutive integers, it is a multiple of 2. Therefore, it can be expressed as (3k + 1)(3k + 2) = 2l, and (p + 1)(p + 2)(p + 3) = 24l(2k + 1). As p, p + 1, p + 2, p + 3, p + 4 are five consecutive integers, one of them is a multiple of 5. If we let p = 5, then p + 4 = 9, which is not a prime number, leading to p + 4 not being prime, hence p > 5, so p, p + 4 are prime numbers greater than 5, hence not multiples of 5. Therefore, one of p + 1, p + 2, p + 3 is a multiple of 5. Therefore, (p + 1)(p + 2)(p + 3) is a multiple of 5. By 2 and 3, (p + 1)(p + 2)(p + 3) is a multiple of 24, hence a multiple of 120.'

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'Among the natural numbers less than 30, there are 15 multiples of 2, 7 multiples of 2^2, 3 multiples of 2^3, and 1 multiple of 2^4. Therefore, the number of prime factor 2 in the prime factorization of 30! is'

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'Find all prime numbers $k$ such that $k^{2}+2$ is a prime number, and prove that there are no other cases.'

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'(1) Find the smallest positive integer n such that n! / 1024 is an integer.'

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'When there are two pairs of dice with only two equal faces each, the only case where the product of two different numbers between 1 and 6 becomes a perfect square is 2^2=1×4, so the sets that satisfy this condition are {1,2,2} and {1,1,4},{2,2,4} and {1,4,4}, in this case k=4,16, leading to k=4,10,15,16,40,90,120'

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'Among natural numbers below 125, there are 25 multiples of 5, 5 multiples of 5^2, and 1 multiple of 5^3. Therefore, the number of prime factor 5 in the prime factorization of 125! is'

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'Prove that if two natural numbers a and b are coprime, then a+b and a*b are also coprime.'

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'Find all combinations of numbers from 0 to 5 where the sum of their digits is a multiple of 3.'

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'Prove that if 49 is a prime number, then $p^{4}+14$ is not a prime number.'

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"The prime factorization of a composite number is unique, except for the order of the factors. Let's prove the uniqueness of prime factorization using the theorem above. Proof: Suppose the prime factorization of composite number a is represented in two different ways."

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'Explain the method to determine whether an integer N is a prime number. For example, check if 257 is a prime number.'

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'For any natural number n greater than 2, let T(n) be the sum of all positive divisors of n (excluding n itself). Find the value of T(120).'

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'Prime number problem\nLet n be a natural number. Prove that the only case where n, n+2, and n+4 are all prime numbers is when n=3.'

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'Key Example 87 | Proof problem about the equation a^2+b^2=c^2\n\nLet a, b, c be natural numbers that do not have any common factors other than 1. When a, b, c satisfy the equation a^2+b^2=c^2, prove the following:\n(1) One of a, b is even and the other is odd.\n(2) If a is odd, then b is a multiple of 4.\n(3) At least one of a, b is a multiple of 3.'

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'Important Example 83 Number of relatively prime natural numbers'

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Prove the following proposition.
(2) If $m n$ is odd, then both $m$ and $n$ are odd.

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For the following cases (1) to (5), find the maximum and minimum values of the quadratic function $y = x^2 - 2ax + a$ in the interval $1 \leqq x \leqq 2$, assuming that a is a constant.
(1) a < 1
(2) 1 \leqq a < \frac{3}{2}
(3) a = \frac{3}{2}
(4) \frac{3}{2} < a \leqq 2
(5) a > 2

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1. Let the minimum value of the quadratic function $y=x^{2}+2 b x+6+2 b$ in the cubic equation $41^{3} x$ be $m$.
(1) Express $m$ in terms of $b$.
(2) Determine the maximum value of $m$ and the corresponding value of $b$ as $b$ varies.

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Basic Exercise 76 Let z=1+i. (2) Find the smallest natural number n such that z^n is a purely imaginary number.