Prove: $\frac{n^5}5 + \frac{n^4}2 + \frac{n^3}3 - \frac n {30}$ is an integer for $n \ge 0$ I am attempting to prove the following problem: 

Prove that $\frac{n^5}5 + \frac{n^4}2 + \frac{n^3}3 - \frac n {30}$ is an integer for all integers $n = 0,1,2,...$

I attempted to solve it by induction, but when proving for $n= x+1$ the algebra gets very messy very fast. I was wondering if this is the only way or if there is a quicker way to prove this. I guess I am a little unsure as to how to prove something is an integer. 
I also noticed that letting $f(x) = \frac{x^5}5 + \frac{x^4}2 + \frac{x^3}3 - \frac x{30}$ and deriving $f(x)$ yields a fairly clean result, but I don't know if this helps me at all. Any help would be great.

Joe, since your original question was to prove that it is an integer:
An extension of Sivaram Ambikasaran's answer:
First claim that 
$\frac{n^5}{5} + \frac{n^4}{2} + \frac{n^3}{3} - \frac{n}{30} = 1^4 + 2^4 +\cdots + n^4$
And show your claim by induction. (This should be straightforward)
Once your claim is proven, the given expression $\frac{n^5}{5} + \frac{n^4}{2} + \frac{n^3}{3} - \frac{n}{30} $ is the sum $1^4 + 2^4 +\cdots + n^4$ for every $n$, which means it is an integer.  Because $1^4 + 2^4 +\cdots + n^4$ for every $n$ is an integer.
