Question

Prove that for any integer n, at least one of the integers n, n+2, n+4 is divisible by 3 .

Answer 1

I don't recall how to organize a proof of this type.

But you might take it one case at a time...

If n is divisible by 3, then you're done.
If n is not divisible by 3, then it must either be 1 greater or 2 greater than a number that is divisible by 3 (i.e., if it was 3 greater than a number divisible by 3, it would also be divisible by 3).

If n is one greater, then n + 2 is three greater than the number divisible by 3, therefore n+2 is also divisible by 3.

If n is two greater than a number divisible by 3, then n + 4 will make it 6 greater than that number, making n + 4 divisible by 3.

Answer 2

That's not a proof, but maybe you could use that logic to formalize in a proof format.

Answer 3

ok thks i'll try

Answer 4

you could use another variable to help.

If n is not divisible by 3, then if m is divisible by 3, n must be m + 1 or m + 2.

Answer 5

so n + 2 = (m + 1) + 2 = m + 3... and since m is divisible by 3, so is m+3

Answer 6

is this make sense?
n+(n+2)+(n+4) = 3n+6 , but (3n+6)/3 = n+2
is that sufficient proof?