2018 AMC 10B Problems/Problem 23
23. How many ordered pairs of positive integers satisfy the equation where denotes the greatest common divisor of and , and denotes their least common multiple?
Let , and . Therefore, . Thus, the equation becomes
Using Simon's Favorite Factoring Trick, we rewrite this equation as
Since and , we have and , or and . This gives us the solutions and , which can be translated back to two solution for and . Thus, the answer is . (awesomeag)