1. Supposea, b,andpare positive integers andpis prime.

Prove that ifp|ab,then either p|a or p|b.

I think the easiest is to use proof by contradiction. Suppose it is not true that either p|a or p|b.

Then this means p does not divide a AND p does not divide b.

But if this is the case, then p must not divide ab as well.

However, a more rigorous proof is go by using greatest common divisor. But I could not finish it.

Let d be the greatest common divisor of a and p. Since p is a prime number, then d must be either 1 or p.

If d = p, it follows by the definition of greatest common divisor that p|a. If d = 1,

it follows that p does not divide a. By assumption p|ab.

It should be reasonable to say that if p does not divide b, then p does not divide ab.

But this seems I have returned to proof by contradiction.

2. Supposeand

are two nondecresing sequences of prime numbers, and

and

Prove that j = k and for

I went by induction on j and showing the base case that

which is not possible if k > 1 due to the fact that p is prime.

As for the inductive step, I went by assuming j = k and for for So,

If for another sequence must be equal , and the fact that , there must be something more to , which is . The inductive hypothesis, j = k and for for and the equality should ensure .

However, the right answer seems to be more rigorous but I think mine should be enough. What's wrong with my argument?