Proof by contrapositive; if (m^2+n^2) div by 4, then m,n are even numbers

[jeszo]

Homework Statement

Let m and n be two integers. Prove that if m^2 + n^2 is divisible by 4, then both m and n are even numbers

Homework Equations

The Attempt at a Solution

Proof by Contrapositive. Assume m, n are odd numbers, showing that m^2 + n^2 is not divisible by 4.

let:
m= 2a + 1 (a,b are integers)
n=2b+1

m^2+n^2 = (2a+1)^2 + (2b+1)^2 = 4a^2 +4a + 1 + 4b^2 + 4b +1

let: 4(a^2 + a + b^2 + b) = 4q (q an integer)

m^2 + n^2 = 4q + 2

with 4q + 2 not divisible by 4 since 4 divides 4q + 2 with a remainder of 2.
==> if m and n are odd numbers, m^2 + n^2 is not divisible 4, which by contrapositive reasoning proves that if m^2 + n^2 is divisible by 4, then m and n are odd numbers.

----------------------------------

I finished my proof there, but had a note underneath that said that it was incomplete, since I was missing "other cases". Does this mean I have to also show that m and n can't be of opposite parity either? I've been under the impression that the negation of "m and n are even numbers" was "m and n are odd numbers". Is this logic wrong?

 

[daveb]

Thenegation of "both m and n are even" is "at least 1 of m or n is odd".