Two positive integer with prime number

Question

Let $a, b$ be distinct positive integers. Prove that there exists a prime $p$ such that when dividing both $a$ and $b$ by $p$, the remainder of $a$ is less than the remainder of $b$.

How can i solve this?

Answer 1

Just take a prime dividing $a$ but not $b$.

Answer 2

Just pick $p$ such that $a<b<p$