1. The problem statement, all variables and given/known data Just as the title said, I need to prove: For any integer n, n2 - 2 is not divisible by 4 by the method of proof by contradiction. 2. Relevant equations (Relevant by division into cases) Even numbers = 2k for some integer k Odd numbers = 2m+1 for some integer m 3. The attempt at a solution 1. Suppose not 2. For any integer n, n2 -2 is divisible by 4 3. n is either even or odd 4. Case 1 - n is even 5. n=2k for some integer k 6. n2 -2=(2k)2 -2 =4*k2 -2 =4 (k2 - 2/4) At this point I know where I need to be, just don't know how to justify that I got there. What I'm basically looking for is the opposite of the closure property - some way to prove that the sum of k^2 and 2/4 is not an integer. As far as I can tell, from there, I can safely state that n^2 -2 is not divisible by 4, do the same thing for odd, and conclude that there is no integer for which n^2-2 is divisible by 4. But how do I reach the fact that k^2-2/4 isn't an integer?