Looking at the "Square of an Odd" puzzle that asks to prove that the square of an odd number is always 1 more than a multiple of 8, a professor gave this four parts proof: "All odd numbers are of the form 8K+1, 8K+3, 8K+5 or 8K+7. Squaring these numbers produces 8M+1, 8M+9, 8M+25 or 8M+49, which are all of the form 8N+1. QED"

Another professor came by, and gave a single line proof. Can you manage it?

Note: no one who answered the original problem produced either the four parts solution, or the single line one.

(In reply to my way by Ady TZIDON)

11 years on:

Because (2n+1)^2-(2n-1)^2 = 8n, all odd numbers must be worth the same as 1, mod 8, namely 1.

Edited on June 24, 2016, 11:50 pm

Posted by broll on 2016-06-24 23:39:48