Question

Write the reminder obtained when $1!+2!+3!+...+200!$ is divided by $14$

Answer 1

$1!+2!+3!+4!+5!+6!+7!.....+200!=1+2+6+24+120+720+5040+......+Integer$

$=(1+2+6+24+120+720)+(14\times 360)+(14\times Integer)$

$=\left(873\right)+14\times (360+Integers)$

$=(5+868)+(14\times Integers)$

$=(5+62\times 14)+(14\times Integers)$

$=5+(14\times 62)+(14\times Integers)$

$=5+14\times (62+Integers)$

$=5+14\times Integers$

$\therefore$ Remainder $=5$