The problem:

(I will presume that d and n are positive integers.) This is one of those "duh!" thoughts but I am messing up the proof somehow. I'd appreciate it is someone could take a look at it.Prove that if d divides n then divides .

First: If d|n then n = kd for some positive integer k. Using their decomposition into powers of primes, let

and

Then

Now, we have that

and

So finally we get

or

Now, I thought that simplified expression to be kinda cool. But as I was thinking about it I checked:

So I've done something incorrectly. (Though I suspect the general structure of the proof is sound.)

Any thoughts?

-Dan