what is the remainder left after dividing 1!+2!+3!...................+100! by 7?

Question

please provide the total process

Answer 1

satwik had the idea
recall that
1! = 1
2! = 1x2 = 2
3! = 1x2x3 = 6
.....
7! = 1x2x3x4x5x6x7 = not necessary since we can see that it has 7 as a factor. so are 8!, 9!, 10! .....

this will cut our solution to 1! + 2! + 3! + 4! + 5! + 6!
but consider that
5! + 6! = 5! + 5!(6)
= 5! (1+6)
= 5! (7) ---which means 5! + 6! is divisible by 7

again this will shorten our solution to just 1! + 2! +3! + 4!

1! = 1 -----divide by 7 gives remainder 1
2! = 2 -----divide by 7 gives remainder 2
3! = 6 -----divide by 7 gives remainder 6
4! = 24 ---divide by 7 gives remainder 3

add these remainders and divide the sum by 7 you'll get 5.

so remainder is 5. //answer

Answer 2

Everything 7! and beyond is divisible by 7, so we only really need to concern ourselves with 1! + 2! + 3! + 4! + 5! + 6!

Dividing 7 into 1! = 1 gives 0 with a remainder of 1.
Dividing 7 into 2! = 2 gives 0 with a remainder of 2.
Dividing 7 into 3! = 6 gives 0 with a remainder of 6.
Dividing 7 into 4! = 24 gives 3 with a remainder of 3.
Dividing 7 into 5! = 120 gives 17 with a remainder of 1.
Dividing 7 into 6! = 720 gives 102 with a remainder of 6.

Adding up the remainders, I get 1+2+6+3+1+6 = 19, which if divided by 7 yields 2 with a remainder of 5.

So the remainder is 5.

Or you could have just added

1!+2!+3!+4!+5!+6! = 873, which if divided by 7 you get 124 with a remainder of 5.

Answer 3

let us break the series
(1!+2!+3!+4!+5!+6!)+(7!+8!+..................)
let the first part be x and next part be y

in y,every term contains 7 .
so it is divisible by 7 leaving no remainder.

consider x
(1!+2!+3!+4!+5!+6!)=1+2+6+24+120+720
=873
=868+5
here 868 is multiple of 7
therefore x leaves a remainder when divided by 7

1!+2!+3!+4!+5!+6!+7!+8!+.................. leaves a remainder 5 when divided by 7

Answer 4

1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! + 9! + 10 ! / 7 =

(1x1) + (1x2) + (1x2x3) + (1x2x3x4) + (1x2x3x4x5) + (1x2x3x4x5x6) + (1x2x3x4x5x6x7) + (1x2x3x4x5x6x7x8) + (1x2x3x4x5x6x7x8x9) + (1x2x3x4x5x6x7x8x9x10) / 7 =

1 + 2 + 6 + 24 + 120 + 720 + 5040 + 40320 + 362880 + 3628800 / 7 = 4,037,913 / 7 =576,844.71

Answer 5

There is no remainder. The series of sums of factorials diverges and so does the series formed by division by 7.

Answer 6

why dont you just write it all out?

wow blah b is pretty clever