TITLE: $16!$ without calculator
QUESTION [9 upvotes]: Scrolling through old discrete mathematics exams I have came across this "choose correct answer" question:

$16!$ is:

a). $20 \; 922 \; 789 \; 888 \; 000$
b). $18 \; 122 \; 471 \; 235 \; 500$
c). $17 \; 223 \; 258 \; 843 \; 600$


Would you show me how your thinking process of solving this problem would look like?
The ultimate goal is to find the correct answer; how you get to it does not matter, except that you have to invest only a reasonable amount of time, and calculators or other devices are not allowed.

REPLY [3 votes]: \begin{align}16!&=1\times2\times3\times4\times5\times6\times7\times8\times9\times10\times11\times12\times13\times14\times15\times16
     \\&=(2\times5\times10)\times3\times4\times6\times7\times8\times9\times11\times12\times13\times14\times15\times16
     \\&=100\times(3\times4\times6\times7\times8\times9\times11\times12\times13\times14\times15\times16)\end{align}
So the final answer will have $2$ zeros at the end. If you want the first digit after the zeros, you can multiply the remaining numbers modulo $10$. 
\begin{align}&3\times4\times6\times7\times8\times9\times11\times12\times13\times14\times15\times16 \pmod {10}
\\&= 3\times4\times6\times7\times8\times9\times1\times2\times3\times4\times5\times6 \pmod {10}
\\&= 2\times6\times7\times8\times9\times2\times3\times4\times5\times6 \pmod {10}
\\&=  2\times7\times8\times9\times2\times3\times4\times5\times6 \pmod {10}
\\&= 4\times8\times9\times2\times3\times4\times5\times6 \pmod {10}
\\&= 2\times9\times2\times3\times4\times5\times6 \pmod {10}
\\&= 8\times2\times3\times4\times5\times6 \pmod {10}
\\&= 6 \times3\times4\times5\times6 \pmod {10}
\\&= 8\times4\times5\times6 \pmod {10}
\\&= 2\times5\times6 \pmod {10}
\\&= 60 \pmod {10}
\\&= 0 \pmod {10}
\end{align}
So the last 3 digits of $16!$ are $000$ and the answer must be a).