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.