Question

prove that if p is prime and ab=0 in Zp then a=0 or b=0

Answer 1

Well, if p were not prime, then there would integers r and s such that
rs = p

Hence if p were not prime and ab = 0, then it would be possible that
ab = np = 0 (mod p)
and so it could be the case that a and b are not 0.

Now, turn this argument around for your case.

Answer 2

I'm as far as saying that p | b or p | a but im not sure how I can go from that to show that one has to be zero

Answer 3

No, if both a and b are not zero, it must be that
a | p or b | p
and ....

Answer 4

then a and b would be factors of p?

Answer 5

...but this leads to a contradiction as p is prime ...

Answer 6

yeah I've gotten there, but I can't close this out

Answer 7

Write out the whole argument.

Answer 8

wait, can I say that because of this contradiction I already know that they both cannot be zero?

Answer 9

Here, try this:

Suppose ab = 0 (mod p) and both a and b are not zero. We want to show this leads to a contradiction.

As ab = 0 (mod p), then ab = np for some n. Now as p is prime, it must be that p is a factor of either a or b (as all integers have unique prime factorizations)

But if a or b has p as a factor, it must necessarily be the case that
a = 0 (mod p) or b = 0 (mod p)