Prove the following statements in Description below.

Question

Prove:

a. The sum of an integer and its cube is even.

b. An integer n is even if and only if n²+37 is odd.

Answer 1

(a)

n + n^3 = n(1 + n^2)

suppose n is odd...

then n=2t+1

(2t+1)(1 + (2t+1)^2) =

(2t+1)(1 + 4t^2 + 4t + 1) =

(2t+1)(4t^2 + 4t+ 2) =

2(2t+1)(2t^2+2t+1) which is divisible by 2

on the other hand, if n is even, then n=2t

(2t)(1 + (2t)^2) which is clearly divisible by 2

(b) ---->

given : N is even.

Then n^2 is even which makes n^2+37 odd

<--

given n^2+37 is odd.

Suppose n is odd.

Then n^2 is odd which makes n^2+37 even, a contradiction.