# Digit in units place of $1!+2!+\cdots+99!$

There isn't much I can add to the question description to expand upon the title. I came across this in a multiple choice test. The options were $3$, $0$, $1$ and $7$. I am absolutely stumped. Any pointers? By the way I ran a computer program and the answer is $3$. The rest of the test was quite easy, apparently I am missing something obvious.

• What is the units digit of $5!$? – Erick Wong Feb 24 '14 at 8:56

Hint: If $n\geq 5$, what is the last digit of $n!$ ?
you need to add only $1!+2!+3!+4!$ which results in last digit as $3$, since number more than $5!$ or $5!$ itself, will have last digit as $0$, due to having a factor $10$.
Notice that: $$1!=1, \\ 2!=2, \\ 3!=6, \\ 4!=24, \\ 5!=120, \\ 6!=6 \times 5!=6 \times 120=720,$$ note that for any $n \geq 5$, $n!$ ended with zero, so it only suffices to sum up $1!+2!+3!+4!$ which is ended by $3$.