Question

The remainder when $1!+2!+3!+...95!$ is divided by 15 is

• A. $3$
• B. $5$
• C. $8$
• D. $12$

Answer 1

The correct option is A $3$

Note that $5!=1\cdot 2\cdot 3\cdot 4\cdot 5$ is divisible by 15, and hence for any number, $n\ge 5$, we have $n!=1\cdot 2\cdot 3\cdot 4\cdot 5\cdots n$ is divisible by 15.
Therefore the remainder when $1!+2!+\cdots +95!$ is divided by 15 is same as the remainder when 15 divides $1!+2!+3!+4!$.
Now, $1!+2!+3!+4!=1+2+6+24=33$ is divided by 15, then the remainder is 3.