Prove that any integer can written as differences of two numbers with same prime divisors.

Prove that any integer can written as differences of two numbers with same prime divisors.

The problem seems wrong since we can prove that it is impossible for $1$ and any prime bigger than $2$.I need the original problem but not an answer since I want to think on it.Maybe we should prove it for any composite number.

• @DietrichBurde I am sure about the part two numbers with same prime divisors. – Taha Akbari Oct 20 '17 at 15:44
• Odd numbers cannot be written as the difference of two numbers with same prime divisors since $\text{odd}=\text{even}-\text{odd}$ or $\text{odd}=\text{odd}-\text{even}$. Even numbers can be written as the difference of two numbers with same prime divisors since $2m=4m-2m$. – mathlove Oct 20 '17 at 17:07

This statement cannot hold for all composite numbers. A simple example is $n=9$. Actually, $n$ is a difference of two numbers with same prime divisors if and only if $n$ is even - see mathlove's argument above.