Find the sum of the multiples of $3$ and $5$ below $709$? I just cant figure this question out:
Find the sum of the multiples of $3$ or $5$ under $709$
For example, if we list all the natural numbers below $10$ that are multiples of $3$ or $5$, we get $3$, $5$, $6$ and $9$. The sum of these multiples is $23$.

Let $n_3 = \lfloor \frac{708}{3}\rfloor, \; n_5 = \lfloor \frac{708}{5}\rfloor, \; n_15 = \lfloor \frac{708}{15}\rfloor.\;$ Then using the hints in the comments your sum $S$ is
$$S=3\sum_{k=1}^{n_3}k + 5\sum_{k=1}^{n_5}k-15\sum_{k=1}^{n_{15}}k
=3\frac{n_3(n_3+1)}{2}+5\frac{n_5(n_5+1)}{2}-15\frac{n_{15}(n_{15}+1)}{2}$$
$$=3\frac{236\times 237}{2}+5\frac{141\times 142}{2}-15\frac{47\times 48}{2} = 117033
$$
Note added in proof: After Darth Geek's first differing answer, I verified my $S=117033$ with a small program.
