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?
Solution 1
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
.
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