I couldn't figure out what to call this. All I know is that it is in a Unique Factorization chapter and I only know the very basics at the moment. I'm just trying to get a head start on number theory before I head into the class this Fall. Here is the problem:
a) Verify that (2^5)(9^2) = 2592 (I can do this but I included this to set the context)
b) Is (2^5)(a^b) = 25ab possible for other a, b? (Here 25ab denotes the digits of (2^5)(a^b) and not a product).
So my problem is I came up with the right solution which is yes and no respectively for a and b but I found both answers by inspection. Part a was easy enough but for part b I noticed there are 81 combinations of numbers and cut that combination down to 2^6, 8^2, 5^2, and 4^3 since they are the closest to 9^2 and so the closest to a number between 2500 and 2599. I checked each number and there was no other combination of a and b to make (2^5)(a^b) = 25ab possible.
Is there a way to do this problem aside from inspection? If anyone can give me a small hint I can try to work it out from there.