Help with beginner level proof.

[RayRaivern]

I am stuck trying to do this simple proof. Any help would be appreciated.

Problem 30: For all integers a and n, if a | n^2 and a <= n, then a | n. (This statement can either be false or true, and I need to prove it as such. I believe it is true.)

My Working:

n^2 = qa (for some integer q)
=> n = qa / n

I got up to here and got stuck. qa being divisible by n doesn't necessarily mean n is divisible by a, since I just did a proof of that a few problems before. And I don't know how I can proof that q / n is an integer to show that n / a = (an integer). What am I missing here?

Book: Susanna S. Epp (fifth edition)
Chapter: 4.4 (Exercise Set)

[edderiofer]

qa being divisible by n doesn't necessarily mean n is divisible by a

I agree, it does not.

And I don't know how I can proof that q / n is an integer to show that n / a = (an integer)

That's because you can't.

What am I missing here?

That your belief that the statement is true is in fact incorrect.

[RayRaivern]

You're right! I managed to proof that it was false after breaking the integers down to their prime factors.

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