Question

show that one and only one of n, n + 2, n + 4 is divisible by 3.

Answer 1

if n is divisible by 3 then n+2 leaves remainder 2 ; and n+4 leaves remainder 1 so none is divisible by 3
Simil. if n+2 is divisible by 3 then n leaves remainder 2 and n+4 leaves remainder 2
simil. if n+4 is divisible by 3 then n leaves remainder 1 and n+2 leaves remainder 2

Answer 2

If n is divisible by 3 then n+2 and n+4 will not be divisible by 3 as only n(+/-)  multiple of 3 will be divisible by 3
similarly if n+2 is divisible be 3 then n and n+4 will not be divisible by 3
and....... if n+4 is divisible by 3 then neither n nor n+2 will be divisible by three