Prove that if $ab \equiv 0 \pmod p$, where p is a prime number, then $a \equiv 0 \pmod p$ or $b \equiv 0 \pmod p$

Question

Prove that if $ab \equiv 0 \pmod p$, where p is a prime number, then $a \equiv 0 \pmod p$ or $b \equiv 0 \pmod p$.

All I have right now is that the prime divisibility property may help with the then part of this problem.

What is the "prime divisibility property"? If it's what I think it is, then this is literally just a one-line application of the definition of congruence. — , Feb 23 '16 at 03:30
The prime divisibility property says that if p|(a1a2... ar), then p divides on of the a's. But now that I look at it, I don't think it helps here — Matt, Feb 23 '16 at 03:34
Why do you think it doesn't help? What is a way to say $p | a$ as a congruence? — , Feb 23 '16 at 03:35
hmmm... couldn't you write a= b+km from the definition of congruence? — Matt, Feb 23 '16 at 03:41
No. That's the statement that $a \equiv b \pmod m$ (or modulo $k$). You should review the definition of $\mod p$. — , Feb 23 '16 at 03:41
I was thinking that but I thought it had to be at least a degree 2 to apply. — Matt, Feb 23 '16 at 03:43
What do you mean by "it had to be at least a degree 2 to apply"? There's nothing here that has a degree. — , Feb 23 '16 at 03:45
I just assumed it had to be. I do understand now that you use this to solve the problem — Matt, Feb 23 '16 at 03:46

score 1 · Answer 1 · answered Feb 23 '16 at 03:32

1

The ideal $(p) \subset \mathbb Z$ is prime, thus if $ab \in (p)$, then $a \in (p)$ or $b \in (p)$.

In other words:

$ab \equiv 0 \pmod p \implies ab=pk \implies p|a$ or $p|b \implies a \equiv 0 \pmod p$ or $b \equiv 0 \pmod p$.

answered Feb 23 '16 at 03:32

Maffred

4,166

Just for clarification: what property is used here? I think I know but I'm not 100% sure. – Matt Feb 23 '16 at 03:48
A prime number which divides a product must divide at least one factor. – Maffred Feb 23 '16 at 03:54
This comes from the unique factorization in $\mathbb Z$ into prime factors. – Maffred Feb 23 '16 at 03:55
That is what I thought it was. Thanks so much – Matt Feb 23 '16 at 03:55

score 0 · Answer 2 · edited Dec 20 '24 at 13:36

Remember that in the integers, for a prime $\;p\;\;,\;\;\;p|a\iff a\equiv kp\;,\;\;a,k\in\Bbb Z\;$ , and then $\;a\equiv0\pmod p\iff p|a\iff a\equiv kp\;$ , so by what you wrote in your second comment below your question:

$$ab\equiv 0\pmod p\iff p|ab\iff p|a\;\;\text{or}\;\;p|b\iff $$

$$a\equiv kp\;\;\text{or}\;\;b\equiv mp\;,\;\;k,m\in\Bbb Z\iff a\equiv 0\pmod p\;\;\text{or}\;\;b\equiv 0\pmod p$$

Prove that if $ab \equiv 0 \pmod p$, where p is a prime number, then $a \equiv 0 \pmod p$ or $b \equiv 0 \pmod p$

2 Answers2