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## Numbers and Algebra

### Basic Number Theory - Least Common Multiples and Greatest Common Divisors

#### Q.01

'The greatest common divisor of a quadratic equation A and a cubic equation B is x+1, and the least common multiple is x^4-x^2. Find the polynomials A and B in this case.'

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'Given that the greatest common divisor of three positive integers 40, 56, and n is 8, and the least common multiple is 1400, find the value of n.'

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'Conditions for the existence of integer solutions to a linear Diophantine equation'

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'Find the greatest common divisor and least common multiple of 48, 240, and 360.'

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'Basic Example 125 Greatest Common Divisor (Euclidean Algorithm)\nFind the greatest common divisor of the following two integers using the Euclidean algorithm:\n(1) 143, 319\n(2) 667, 966\n(3) 1829, 2077'

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'Application problem of the greatest common divisor (tiling)'

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'PRACTICE 125\nFind the greatest common divisor of the following two integers using the Euclidean algorithm:\n(1) 504, 651\n(2) 899, 1501\n(3) 2415, 9345'

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'Greatest common divisor (using Euclidean algorithm)'

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'Find the two two-digit natural numbers m and n (m < n) where the greatest common divisor is 10 and the least common multiple is 100.'

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'When the three positive integers are 40, 56, and n, and their greatest common divisor is 8, and their least common multiple is 1400, find the value of n.'

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'87 (1) Greatest common divisor 66, Least common multiple 792\n(2) Greatest common divisor 21, Least common multiple 1260'

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'Find all natural numbers n such that the least common multiple of n and 28 is 980.'

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'Find the greatest common divisor and least common multiple of 24, 120, and 180.'

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'Using Euclidean algorithm to find the greatest common divisor of 105.'

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'Find two natural numbers, where the greatest common factor is 6 and the least common multiple is 432.'

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'Find the greatest common divisor and least common multiple of the following set of integers:\n2) 84, 252, 315'

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'Find the greatest common divisor and least common multiple of 36 and 120.\nDivide by common prime factors, 12 divided by the prime factors on the left.\nThe product of the left prime factors is the greatest common divisor, and the product of the two bottom numbers in the left prime factors is the least common multiple.'

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'Find the two natural numbers whose greatest common divisor is 11, least common multiple is 1320, and sum is 253.'

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'Find the greatest common divisor of the following two integers using Euclidean algorithm:\n105\n(1) 767,221\n(2) 966,667\n(3) 1679,837'

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'Find the greatest common divisor and least common multiple of 8 and 12.'

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'Find two natural numbers such that their greatest common divisor is 6 and least common multiple is 432.'

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'Find the greatest common divisor and least common multiple of 36 and 120.'

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'Find the greatest common divisor and least common multiple of the following pairs of integers. (1) 198, 264 (2) 84, 252, 315'

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'Find two natural numbers where the greatest common divisor is 3, the least common multiple is 210, and the sum is 51.'

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'(1) Find all natural numbers n such that the least common multiple of n and 36 is 720.\n(2) Find all natural numbers n such that the least common multiple of n, 12, and 50 is 1500.'

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'Given that the greatest common divisor of 238 and a natural number n is 14, and the least common multiple is 1904. Find the value of n.'

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"Let A, B be polynomials with greatest common divisor G and least common multiple L. If we divide A, B by G and obtain quotients A', B', what is the relationship between them?"

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'Greatest common divisor and least common multiple of polynomials'

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'Find the greatest common divisor and least common multiple of two numbers: 132 and 792'

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'Find the greatest common divisor and least common multiple of the following set of numbers: 84, 252, 315.'

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'Find the greatest common divisor and least common multiple of the following 26 sets of numbers.'

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"(1) Since the greatest common divisor is 24, a and b can be expressed as a=24 a' and b=24 b', where a' and b' are coprime natural numbers and a'<b'. In this case, the least common multiple of a and b is 24 a' b' which equals 432. Therefore, a' b' equals 18."

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'Euclidean algorithm: Explain the procedure to find the greatest common divisor of two natural numbers a, b.'

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"Given two natural numbers a and b, find their greatest common divisor (GCD) using Euclid's algorithm."

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'Therefore, the least common multiple l of g and m equals g, and g is a multiple of m.'

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'After surveying 100 university students, it was found that 75 students own a personal computer and 80 students own a mobile phone. Let x be the number of students who own both a personal computer and a mobile phone, find the minimum possible value of x. Furthermore, if there are 60 students among the 100 who own a car, let y be the number of students who own all three devices: a personal computer, a mobile phone, and a car. What is the minimum possible value of y?'

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"Properties of greatest common divisor and least common multiple\nLet the greatest common divisor of two natural numbers a, b be g, and the least common multiple be l. If a=g a' and b=g b', then especially, when g=1, ab=l"

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'Practice 93 booklet p. 428\nQuestion 2\nThe greatest common divisor of numbers A and B is denoted by (A, B).\n(1)\n...\nTherefore, the greatest common divisor of 3a + 7b and 2a + 5b is equal to the greatest common divisor of a and b.'

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'Find the greatest common divisor of the following two integers using Euclidean algorithm:\n(1) 504, 651\n(2) 943, 1058\n(3) 4165, 6035'

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"Since the greatest common divisor is 9, a, b can be represented as a=9a', b=9b'. Where a' and b' are coprime natural numbers and the product of a, b is 486, therefore: 81a'b' = 486, which means a'b' = 6"