Use a proof by contraposition to show that If m2 n2 is an o

Use a proof by contraposition to show that If m2 + n2 is an odd integer, then m is odd or n is odd.

Solution

We have to prove by contrapostive that if m²+n² is odd then m is odd or n is odd.

By contrapositive, we have to show that if m and n are even, then m²+n² is even.

Say,

m = 2p

n = 2q

Therefore,

m²+n² = (2p)² + (2q)² = 4p²+4q² = 4(p²+q²)= Even Number.

Therefore, by contrapositive, if m² + n² is an odd integer, then m is odd or n is odd