Is There a Name for This Theorem?

[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?

 

[haruspex]

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).

 

[daniel.e2718]

OP, did you mean to reverse those...?

$b|a \; \rightarrow \; b|a^{2}$

Is certainly true.

 

[AlephZero]

Is certainly true.

But it's hardly worth calling it a theorem, since it's just a special case of $b|a \rightarrow b|ac$.

 

[Benn]

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.

 

[Millennial]

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