homework-6

Question

Answer 1

• A. Using Direct Proof : Let’s assume n = 2x+1 (using definitions of odd numbers). n*n = (2x+1)(2x+1) = 2(2x*x+2x)+1 => n*n = 2(some integer)+1. Therefore n*n is odd.
• B. Using Direct Proof : Let’s assume n = 2x (using definitions of even numbers). n*n = (2x)(2x) = 2(2x*x) => n*n = 2(some integer). Therefore n*n is even.

(x,y are some integers)