Please check argument for 2 questions about prime numbers.
1. Suppose a, b, and p are positive integers and p is prime.
Prove that if p|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. Suppose and
are two nondecresing sequences of prime numbers, 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?