A number theory problem by Paola Ramírez

LaTeX: $\large 1+2+3+\cdots+10^{2016}$

$1+2+3+ \cdots+10^{2016} = \frac{10^{2016} \times (10^{2016}+1)}{2}$

Since $(10^{2016}+1)$ is odd, then it doesn't have a factor which is a power of 2 aside from 1.

$\frac{10^{2016}}{2} = \frac{5^{2016} \times 2^{2016}}{2} = 5^{2016} \times 2^{2015}$

Therefore, 2 appears $\boxed{2015}$ times in the prime factor decomposition of the $1+2+3+ \cdots+10^{2016}$