AIME #7 1987

[stupidityismygam]

Let [unparseable or potentially dangerous latex formula] denote the least common multiple of positive integers [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] Find the number of ordered triples [unparseable or potentially dangerous latex formula] of positive integers for which [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]

[stupidityismygam]

[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

Since [unparseable or potentially dangerous latex formula] the greatest power of 2 and 5 in either [unparseable or potentially dangerous latex formula] is [unparseable or potentially dangerous latex formula]

Since [unparseable or potentially dangerous latex formula] the greatest power of 2 in either [unparseable or potentially dangerous latex formula] is 4, and the greatest power of 5 in either [unparseable or potentially dangerous latex formula] is 3. The same follows for [unparseable or potentially dangerous latex formula].

This means that the greatest power of 2 in [unparseable or potentially dangerous latex formula] must be 4. If [unparseable or potentially dangerous latex formula] then there are 4 choices for the greatest power of 2 in [unparseable or potentially dangerous latex formula], and if [unparseable or potentially dangerous latex formula] then there are 3 more choices (note the case [unparseable or potentially dangerous latex formula] has already been covered). So for the powers of 2, there are [unparseable or potentially dangerous latex formula] choices

Now for the powers of 5. There must be two of [unparseable or potentially dangerous latex formula] with the greatest power of 5 that divides it is 3. So then we have all possible arrangements of (3,3,0) (3,3,1) (3,3,2) and (3,3,3) which is [unparseable or potentially dangerous latex formula]

So there are 10 possibilities for the power 5 and 7 possibilities for the power 2, so [unparseable or potentially dangerous latex formula]