Prove the following statements by contraposition A) For all integers m and n, if

Question

Prove the following statements by contraposition
A) For all integers m and n, if mn is odd then m is odd and n is odd.
B) For all integers n, if n^2 mod 3 = 1 then n mod 3=1 or n mod 3 =2.
Prove the following statements by contraposition
A) For all integers m and n, if mn is odd then m is odd and n is odd.
B) For all integers n, if n^2 mod 3 = 1 then n mod 3=1 or n mod 3 =2.
Prove the following statements by contraposition
A) For all integers m and n, if mn is odd then m is odd and n is odd.
B) For all integers n, if n^2 mod 3 = 1 then n mod 3=1 or n mod 3 =2.

A) For all integers m and n, if mn is odd then m is odd and n is odd.
B) For all integers n, if n^2 mod 3 = 1 then n mod 3=1 or n mod 3 =2.

Explanation / Answer

A) For all integers m and n, if mn is odd then m is odd and n is odd.

The contrapositive is For all integers m and n, if m is not odd or n is not odd then mn is not odd.

Proof: If m is not odd, m is even. Let m = 2k

Then mn = 2k*n which is divisible by 2 and hence not odd.

If n is not odd, n is even. Let n = 2l

Then mn = m*2l which is divisible by 2 and hence not odd.

Thus if m is not odd or n is not odd then mn is not odd.

By contraposition, if mn is odd then m is odd and n is odd.

B) For all integers n, if n2 mod 3 = 1 then n mod 3=1 or n mod 3 =2.

The contraposition is if n mod 3 1 and n mod 3 2 then n2 mod 3 1.

Proof: Since n mod 3 1 and n mod 3 2, n mod 3 = 0, since 0,1 and 2 are the only possible remainders when n is divided by 3.

Since n mod 3 = 0, let n = 3k.

=> n2 = 3kn

Since RHS has a factor 3, LHS must be divisible by 3

=> n2 = 0 mod 3

=> n2 1 mod 3

Therefore, if n mod 3 1 and n mod 3 2 then n2 mod 3 1.

By contraposition, For all integers n, if n2 mod 3 = 1 then n mod 3 = 1 or n mod 3 = 2.

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