Let x be an integer Prove that if 11x5 is odd then x is even

Let x be an integer. Prove that if 11x-5 is odd, then x is even by a) ProofbyContraposition b) ProofbyContradiction.

Solution

a) Proof by Contraposition

Given if 11x-5 is odd, then x is even.

By Contraposition if x is not even then 11x-5 is not odd.

Let x be an integer.

x = 2k + 1

11x = 11(2k + 1)

11x = 22k+11

11x-5 = 22k+11-5

11x-5 = 22k+6

11x-5 = 2(11k+3)

2(11k+3) will be even so 11x-5 will also be even.

means if x is not even then 11x-5 is not odd.

means if 11x-5 is odd, then x is even.

b) Proof by Contradiction----->

Given 11x-5 be an odd integer.

Assume By Way of Contradiction that x is odd.

Then x = 2k+1 for some integer k.

Thus 11x-5 = 11(2k+1)-5 = 22k + 6 = 2(11k+3).

Since 11k+3 is an integer, 2(11k+3) is an even integer. So 11x-5 is also even.

Thus contradicting the fact that 11x-5 is odd, so x must be even.