Biconditional statements with Or

[Ateowa]

Biconditional statements with "Or"

If I have a biconditional statement like this: Let p be an integer other than 0, -1, +1. Prove that p is prime if and only if for each a that exists in Z either (a, p) =1 or p|a.

I know that when you have a biconditional, you have to prove the statement both ways. However, when you solve it the other way, do you have to switch it to an "and," or do you have to do three proofs, one forward, and two backward (One for each case)? I have a homework problem where I have to prove this statement, and going forward is easy but to prove it the other way around, I feel like I need both of those statements to actually prove it.

I'm sorry if this is the wrong place to put it!

 

[_Andreas]

Yes, you have to do one for each case. The "either... or" condition makes it clear that each case taken alone (gcd*(a,p)=1; p|a) is sufficient for p to be prime.

* I guess.