Next step to take to reach the contradiction?

Question

This problem is from Discrete Math and its Applications

I am trying to use proof by contradiction to do this problem, proof by contradiction as described by the book

Here is my work so far for this problem

What I did was i formed a proposition p that represents the problem at hand - there is no positive integer n such that n^2 + n^3 = 100. This is a proposition because it has a true/false value - there either is this positive integer or there isn't. From my understanding of proof by contradiction, I am supposed to assume that ~p(not p) is true, meaning there is a positive integer n such that n^2 + n^3 = 100 and show that this will lead to a contradiction(proposition that is always false no matter what input values are). If it does lead to a contradiction, the proposition ~(p) is false, meaning p is true. I am tried to do this by first factoring out a n^2, getting n^2(n+1) = 100. What step do i take next to prove this expression is a contradiction(false, no matter what value n is passe in)? I can't use the quadratic formula because there is a n cubed in this problem

Answer 1

You are near.

Hint: Use that $n$ is positive integer. What can be $n$ if you know that $n^2$ divides $100$?

Answer 2

You are on the right track. Remember that $n$ is an integer, this means that $n$ is either even or odd. If $a*b=100$, and $a, b\in \mathbb{N}$, what do you know about $a$ and $b$ (regarding whether they are even or odd)?