1987 AIME Problems/Problem 7
Problem
Let denote the least common multiple of positive integers and . Find the number of ordered triples of positive integers for which , , and .
Contents
[hide]Solution 1
It's clear that we must have , and for some nonnegative integers . Dealing first with the powers of 2: from the given conditions, , . Thus we must have and at least one of equal to 3. This gives 7 possible triples : and .
Now, for the powers of 5: we have k, n, q(3, 3, 3)(3, 3,3 n)(3, n, 13)(n, 3, 3)$.
Since the [[exponent]]s of 2 and 5 must satisfy these conditions independently, we have a total of$ (Error compiling LaTeX. Unknown error_msg)7 \cdot 10 = 70$ possible valid triples.
Solution 2
and . By looking at the prime factorization of , must have a factor of . If has a factor of , then there are two cases: either (1) or , or (2) one of and has a factor of and the other a factor of . For case 1, the other number will be in the form of , so there are possible such numbers; since this can be either or there are a total of possibilities. For case 2, and are in the form of and , with and (if they were equal to 3, it would overlap with case 1). Thus, there are cases.
If does not have a factor of , then at least one of and must be , and both must have a factor of . Then, there are solutions possible just considering , and a total of possibilities. Multiplying by three, as , there are . Together, that makes solutions for .
See also
1987 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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