## Is There a Name for This Theorem?

Is there a theorem that says when b|a2 → b|a is true for integers a and b?

If so, what is it called?

 Recognitions: Homework Help Science Advisor I hope not, since it isn't generally true. 9|36, but not 9|6. You would need that b is a prime (or at least, has no repeated prime factors).
 OP, did you mean to reverse those...? $b|a \; \rightarrow \; b|a^{2}$ Is certainly true.

Recognitions:

## Is There a Name for This Theorem?

 Quote by daniel.e2718 Is certainly true.
But it's hardly worth calling it a theorem, since it's just a special case of ##b|a \rightarrow b|ac##.

 Quote by Dschumanji Is there a theorem that says when b|a2 → b|a is true for integers a and b? If so, what is it called?
That is true whenever b is prime. You can prove it by using euclid's lemma.

Let b be prime. Suppose b|a2. Then b|aa, and, by euclid's lemma, b|a or b|a. Hence b|a.

 The statement holds true whenever $|\mu(b)|=1$.