Pairty of integers

Number Theory Level 2

$\large x = (a-2)^4 + (a+3)^3$

Integers $a$ and $x$ satisfy the equation above. Which statement is always true?

x

is odd

a

is even

x

is even

a

is odd

1 solution

João Areias
Nov 22, 2018

Suppose $a$ is even, then: $a-2$ is even so $(a-2)^2$ is even, $a+3$ is odd so $(a+3)^2$ is odd, so x is odd.

Now suppose $a$ is odd, then: $a-2$ is odd so $(a-2)^2$ is odd, $a+3$ is even so $(a+3)^2$ is even, so x is odd.