# I need help on manipulating this expression:

Assume that $(4k + 3) ^ 2 - (4k + 3)$ is not divisible by 4. If this is true, prove that $(4(k+1) + 3) ^ 2 - (4(k+1) + 3)$ is not divisible by 4.

I need to prove this for my induction problem, and I've tried manipulating the second expression as much as I can but haven't arrived at a situation where I can use the first expression. Any help is appreciated, could anyone show me how to do this? Thank you.

Let $a=4k+3$. Then $$b=(4(k+1) + 3) ^ 2 - (4(k+1) + 3)=(a+4)^2-(a+4)=(a^2-a)+4(2a+3)$$ So, $b$ is a multiple of $4$ iff $a^2-a$ is a multiple of $4$, which it is not, by hypothesis.