For a three-digit number $xyz$, where x, y, and z are the digits of the number, $f(xyz)=5^x 2^y 3^z$. If $f(abc)=3f(def)$, what is the value of $abc-def$?

This one took me a while. The key is realizing the function depends on the digits, not the number itself, so f(abc)=3f(def) means the ratio of the digit-products is 3. I had to think about which digit changing by 1 would multiply the expression by 3.

Wait, is the answer just the difference in the digits? That seems too simple for a 700-level question.

Nice question. I like that it forces you to compare the prime factorizations. 3 is already in the function, so you need z to increase by 1, but then you have to check the other digits can't change.

Got E. The exponents make it clear: to multiply by 3 you need 3^1 more, so z goes up by 1. Then abc - def = 1 if only the units digit changes? Actually I need to be careful about place value.