Question

What is the remainder when $1!+2!+3!....2014!$ is divided by $21$ ?

• A. $7$
• B. $5$
• C. $12$
• D. $17$

Answer 1

The correct option is C $12$

$N\Rightarrow 1!+2!+3!.....+2014!$

To find reminder when N is divide by 21

All the factorials bayed and equal to 7

will be divisible by 21 & Remainder will be 0

(becoz they contain 3 & 7 in their factorials)

$(21=3\times 7)$

$\therefore$ our Question how is

$\frac{1!+2!+3!+4!+5!+6!}{21}$

$\Rightarrow \frac{1+2+6+24+120+720}{21}$

$=\frac{873}{21}$

and Remainder $\Rightarrow 12$

(option C is correct)