Prove by contraposition.?

Question

if 8 does not divide x^2-1, then x is even.

Answer 1

You negate both assertions, but reverse their positions: if A then B becomes if not B then not A

So,

if x is odd then 8 divides x^2 - 1

Since x is odd there is a number k such that x = 2k + 1.

x^2 = (2k + 1)^2 = 4k^2 + 4k + 1

So

x^2 - 1 = 4k^2 + 4k + 1 - 1 = 4k^2 + 4k = 4k(k+1)

Now then, if k is even, the k = 2r so that we have 8r(2r + 1) which is divisible by 8.

If k is odd then k = 2r + 1 for some r, but then we have

4(2r + 1)(2r + 2) = 4(2r + 1)(2)(r + 1) = 8(2r + 1)(r + 1) which is also divisible by 8.

HTH

Charles