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.*

Then, the solution would be as follows:

Line 1:

Every odd numbers are of the form 2M-1 (for positive M), the square of which is: 4M(M- 1) + 1.

Line 2:

Since M(M-1) must always be even (or, zero), it follows that the square of any odd number must be precisely 1 more than a multiple of 8.

*Edited on ***March 24, 2010, 4:25 pm**