I am proving that $$5n^3 + 7n^5 \equiv 0 \pmod{12}$$ It would suffice to show $$n^3 \equiv n^5 \pmod{12}$$ How would I go about doing that?
I suppose I could just go through each $n \equiv r \pmod{12}$ with $r$ from $1$ to $11$ and show that $n^3 \equiv n^5 \pmod{12}$ for each, but that would be tedious. Surely there's a better way.
We need to show $$12|n^5-n^3$$ for each $n$. Factor this as $$n^3(n-1)(n+1).$$ This quantity will always be divisible by 3 (why?) and will always be divisible by 4 (why?) so it is divisible by 12.
Hint: If I have three consecutive integers one is divisible by 3.
Hint: If $n$ is even then we are done. If $n$ is odd both $n+1$ and $n-1$ are divisible by 2.
Remark: By this exact reasoning we actually get $24|n^5-n^3$ for each $n$.
$$\begin{align}n^5-n^3&=(n-2)(n-1)n(n+1)(n+2)-4n^3+4n\\\\ &=(n-2)(n-1)n(n+1)(n+2)-4(n-1)n(n+1)\end{align}$$
The first expression is a product of 5 consecutive numbers, hence is divisible by $5!$ i.e., by 120. Similarly, the second expression is divisible by $4\cdot (3!)$ i.e., by 24.
Also, $n^5-n^3 = n^2(n-1)n(n+1)$. Clearly, a product of 3 consecutive numbers $(n-1)n(n+1)$ is divisible by 3. Again, $n^5-n^3 = n^3(n^2-1)$.
If $n$ is even, the R.H.S. is divisible by 8. If $n$ is odd (equals to $2m+1$, say), then $$n^2-1 = (2m+1)^2 - 1 = 8m(m+1)/2$$ i.e, is divisible by 8 (See If $n$ is an odd natural number, then $8$ divides $n^{2}-1$).
It's just case $\rm\ p^j = 2^2,\ q^k = 3,\ d = e = 2\ $ of this simple generalization of Euler's little theorem
THEOREM $\ $ For primes $\rm\:p \ne q\:,\:$ naturals $\rm\:e\:$ and $\rm\ j,\ k \:\le\: d\ $
$$\rm\quad\quad\ \phi(p^j),\ \phi(q^k)\ |\ e\ \ \Rightarrow\ \ p^j\ q^k\ |\ n^d\ (n^e - 1)\ \ \ \forall\ n\in \mathbb N $$
Proof $\ $ If $\rm\ p\ |\ n\ $ then $\rm\ p^j\ |\ n^d\ $ by $\rm\ j\le d\:.\:$ Else $\rm\:n\:$ is coprime to $\rm\: p\:,\:$ so by Euler's little theorem we have: $\ $ mod $\rm\ p^j\:,\ \ n^{\phi(p^j)}\equiv 1\ \Rightarrow\ n^e\equiv 1\ $ by $\rm\ \phi(p^j)\ |\ e\:.\ $ Thus $\rm\ n^d\ (n^e - 1)\ $ is divisible by $\rm\ p^j\ $ and, similarly it is divisible by $\rm\ q^k\:,\ $ hence it is also divisible by their lcm = product. $\quad$ QED
In fact for $\rm\ p = 2,\ j > 2\ $ we can use $\rm\ \phi(2^j)/2\ $ vs. $\rm\ \phi(2^j)\ $ because $\rm\ \mathbb Z/2^j\ $ has multiplicative group $\rm\ C(2)\times C(2^{j-2})\ $ for $\rm\ j> 2\:$. Thus, in fact, $\rm\ 24\ |\ n^3\ (n^2 - 1)\:.$ For more see my post on the Fermat-Euler-Carmichael theorem.
$$\begin{align}n^5-n^3&=6\binom{n+1}{3}n^2\\&=4\binom{n+1}{2}\binom{n}{2}n\end{align}$$
Since $6,4\mid n^5-n^3$ we also have that $\text{lcm}(4,6)=12\mid n^5-n^3$
