# Is There a Name for This Theorem?

1. Jul 14, 2012

### 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?

2. Jul 15, 2012

### 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).

3. Jul 15, 2012

### daniel.e2718

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

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

Is certainly true.

Last edited: Jul 15, 2012
4. Jul 15, 2012

### AlephZero

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

5. Jul 16, 2012

### Benn

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.

6. Jul 16, 2012

### Millennial

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