The setup is

So the problem reads:

Now, we know that P, E, and M are all less than 5 since PEEM is a number in base 5.

So we can start by postulating P = 1:

M is at most 4, so we need E to be at least 3. (And, as it turns out, less than 4.) So try E = 3:

M = 2.

So PEEM = 1332 works.

I can find no other solutions.

-Dan