TITLE: Nuking the Mosquito — ridiculously complicated ways to achieve very simple results
QUESTION [66 upvotes]: Here is a toned down example of what I'm looking for:
Integration by solving for the unknown integral of $f(x)=x$:
$$\int x \, dx=x^2-\int x \, dx$$
$$2\int x \, dx=x^2$$
$$\int x \, dx=\frac{x^2}{2}$$
Can anyone think of any more examples?
P.S. This question was inspired by a question on MathOverflow that I found out about here. This question is meant to be more general, accepting things like solving integrals and using complex numbers to evaluate simple problems.

REPLY [3 votes]: $n!$ is not a power of a prime for $n\geq 3$.
Boring proof: $n!$ is divisible by two distinct primes $2$ and $3$.
Cool proof: For $n\geq 5$, the only normal subgroups of $S_n$ are $\{1\},A_n$ and $S_n$. Since $A_n,S_n$ are not commutative, it follows that the center $Z(S_n)$ is trivial. On the other hand, if $|S_n|=n!$ was a power of a prime, then $S_n$ would be a (nontrivial) $p$-group, and such always has a nontrivial center, which is a contradiction.