Question

Proof that and give two example in words (explain) of the following proposition If x^2 -8x +7 is even then x is odd

Answer 1

=>  x^2 - 8x + 7
=>  x^2 - 7x - x + 7
=>  x(x - 7) - 1(x - 7)
=>  (x - 7) * (x - 1)

we are given that (x - 7) * (x - 1) is even

then (x - 7) must be even and (x - 1) must be even

given (x - 7) is even, then x is odd
given (x - 1) is even, then x is odd

so, if x^2 - 8x + 7 is even then x is odd.

Examples:
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1)
take an odd value for x. x = 1
then x^2 - 8x + 7 is 0 which is even.

2)
take an odd value for x. x = 3
then x^2 - 8x + 7 is -8 which is even.

so, given x^2 - 8x + 7 is even then x is odd.