1987 AIME Problems/Problem 7

Problem

Let $\displaystyle [r,s]$ denote the least common multiple of positive integers $\displaystyle r$ and $\displaystyle s$ . Find the number of ordered triples $\displaystyle (a,b,c)$ of positive integers for which $\displaystyle [a,b] = 1000$ , $\displaystyle [b,c] = 2000$ , and $\displaystyle [c,a] = 2000$ .

Solution

It's clear that we must have $a = 2^j5^k$ , $b = 2^m 5^n$ and $c = 2^p5^q$ for some nonnegative integers $j, k, m, n, p, q$ . Dealing first with the powers of 2: from the given conditions, $\max(j, m) = 3$ , $\max(m, p) = \max(p, j) = 4$ . Thus we must have $p = 4$ and at least one of $m, j$ equal to 3. This gives 7 possible triples $(j, m, p)$ : $(0, 3, 4), (1, 3, 4), (2, 3, 4), (3, 3, 4), (3, 2, 4), (3, 1, 4)$ and $(3, 0, 4)$ .

Now, for the powers of 5: we have $\max(k, n) = \max(n, q) = \max(q, k) = 3$ . Thus, at least two of $k, n, q$ must be equal to 3, and the other can take any value between 0 and 3. This gives us a total of 12 possible triples.

Since the exponents of 2 and 5 must satisfy these conditions independently, we have a total of $7 \cdot 12 = 084$ possible valid triples.

1987 AIME (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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1987 AIME Problems/Problem 7

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