1. The problem statement, all variables and given/known data if n is a natural number and n2 is odd, then n is odd 2. Relevant equations odd numbers: 2k+1, where k is an integer even numbers: 2K, where k is an integer 3. The attempt at a solution ok so take the opposite to be true, or n2 is odd and n is even. Then we would have n2= 2p+1 and n=2k, where p and k are both integers. then take n2= (2k)2= (22)(k2)= 4k2 Then set 2p+1= 4k2→ 4k2-2p=1 2(2k2-p)=1 (2k2-p)=1/2 but since we know 2k2 is an integer because it is just the product of 3 integers, 2*k*k, and we also know p is an integer by definition, we also know that (2k2-p) is an integer because it is the subtraction of two integers. This leaves us with a contradiction since we know (2k2-p) can not equal 1/2, hence proof is complete.