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Determine the number of positive integer pairs (m, n) where GCD (m, n) = 12 and LCM (m, n) = 360.

User Almond
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Final answer:

There are 8 positive integer pairs (m, n) that satisfy GCD(m, n) = 12 and LCM(m, n) = 360.

Step-by-step explanation:

We need to find the number of positive integer pairs (m, n) where GCD(m, n) = 12 and LCM(m, n) = 360. The GCD (Greatest Common Divisor) is the largest number that divides both m and n. The LCM (Least Common Multiple) is the smallest number that is divisible by both m and n.

GCD(m, n) = 12

This means that both m and n are divisible by 12. So, we can write m = 12a and n = 12b, where a and b are positive integers.

LCM(m, n) = 360

Since LCM(m, n) = (m * n) / GCD(m, n), we can substitute the values of m and n and simplify the equation:

360 = (12a * 12b) / 12

360 = 12ab

30 = ab

So, we need to find the number of positive integer pairs (a, b), where the product of a and b is 30.

The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, and 30.

Therefore, there are 8 positive integer pairs (a, b) that satisfy ab = 30.

Since m = 12a and n = 12b, there are 8 positive integer pairs (m, n) that satisfy GCD(m, n) = 12 and LCM(m, n) = 360.

User Willwill
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