1. Relations (double check?)

A binary relation P is defined on Z as follows: For all $\displaystyle m, n \in Z$ m P n <=> exists a prime number p such that p|m and p|n

1) Is the relation reflexive?
2) Is the relation transitive?
3) Is the relation symmetric?

For 1 I'm assuming the answer's right, because if m = n then p|m must mean p|n. For 3 I'm assuming it is because going from $\displaystyle m, n \in Z$ to $\displaystyle n, m \in Z$ results in the same relationship.

What I'm not sure about is 2. I think it is transitive, because if x P y, and y P z, since P means p divides both parameters, then x P z must be valid as well.

I'm not entirely sure, but that's what I have so far. So it would be reflexive, transitive and symmetric?

2. Originally Posted by Open that Hampster!
A binary relation P is defined on Z as follows: For all $\displaystyle m, n \in Z$ m P n <=> exists a prime number p such that p|m and p|n
1) Is the relation reflexive?
2) Is the relation transitive?
3) Is the relation symmetric?
For 1 I'm assuming the answer's right, because if m = n then p|m must mean p|n. For 3 I'm assuming it is because going from $\displaystyle m, n \in Z$ to $\displaystyle n, m \in Z$ results in the same relationship.

What I'm not sure about is 2. I think it is transitive, because if x P y, and y P z, since P means p divides both parameters, then x P z must be valid as well.

I'm not entirely sure, but that's what I have so far. So it would be reflexive, transitive and symmetric?
Is it true that $\displaystyle (2,10)\in\mathcal{P}~\&~(10,25)\in\mathcal{P}~?$
Is it the case that $\displaystyle (2,25)\in\mathcal{P}~?$