Question

What is remainder of 1! + 2! + . . . + 1000!, when divided by 12.

Answer 1

12= 3×4

1! =1

2!=2×1

3!=3×2×1

4!=4×3×2×1

5!=5×4×3×2×1

...

...

...

1000!=1000×999×...×4×3×2×1

Thus,

12/4!, 12/5! ,12/6!......

In other words,

4! congruent to 0 [mod 12]

5!congruent to 0[mod 12]

...

1000! Congruent to 0 [mod 12]

So,

1!+2!+3!+.....1000!is congruent to

1+2×1+3×2×1+0+0+0....+0[mod 12] is congruent to

1+2+6[mod 12] is congruent to

9[mod 12]

Therefore, the remainder is 9.