There exists a number m, which is an element of the positive integers, that for all positive integers n, n+m can be divided by 3. Prove whether this statement is true or false.
The Attempt at a Solution
I ran into a similar question earlier on, which just had the initial part reversed (as in, for all positive integers n there is a positive integer m so that n+m is divisible by 3). I proved that statement by letting m = 2n, and then 3n / 3 = n, which is a positive integer, proving that n+m is divisible. However, I don't understand why reversing the initial condition suddenly makes the entire statement false. Can I not do the same m = 2n idea to prove this statement? Could anyone explain why this is? Thank you in advance.