## Factorial Sum

Here is a sum from MATHCOUNTS:

What are the last two digits in the sum of the factorials of the first 100 positive integers?

From 1! to 4! you can add the units digits, since 5! to ... have 0 in their units place.

From that I get 13, and I carry over the 1 over to the next column and add the tens digits of 1! to 9! since 10! to ... have 0 in their tens and units place.

 So $$\sum_{n=1}^{100} n!$$ $$0!: 01$$ $$1! : 01$$ $$1!+2!: 03$$ $$1! + 2! + 3!: 09$$ $$1!+2!+3!+4!: 33$$ $$1!+2!+3!+4!+5!: 53$$ Can you see a pattern? Go up to $$9!$$ because $$\sum_{n=1}^{10} n!$$ has the last two digits $$00$$. Therefore $$\sum_{n=1}^{\19} n!$$ has last two digits $$13$$ as does $$\sum_{n=1}^{100} n!$$

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 Quote by courtrigrad So $$\sum_{n=1}^{100} n!$$ $$0!: 01$$ $$1! : 01$$ $$1!+2!: 03$$ $$1! + 2! + 3!: 09$$ $$1!+2!+3!+4!: 33$$ $$1!+2!+3!+4!+5!: 53$$ Can you see a pattern? Go up to $$9!$$ because $$\sum_{n=1}^{10} n!$$ has the last two digits $$00$$. Therefore $$\sum_{n=1}^{\19} n!$$ has last two digits $$13$$ as does $$\sum_{n=1}^{100} n!$$
I presume you meant to say that 10! has last two digits 00, not $$\sum_{n=1}^{100} n!$$.

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## Factorial Sum

1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = ...20 (the dots stand for some digits, but I didn't calculate them, since I am only interested in the last two digits)
9! = ...80
10! = ...800

Which numbers in the sum of factorials contribute to the last digit? It's
1! = 1
2! = 2
3! = 6
4! = 24
the other numbers have 0 as their last digit.
Thus, the last digit of our factorial sum is 3 because 1+2+6+24=33

Which numbers in the sum of factorials contribute to the "10" digit (the digit left to the last digit)?
It's

5! = 120
6! = 720
7! = 5040
8! = ...20
9! = ...80

AND the 33 (= sum from 1! to 4!)

I wrote down the relevant numbers again and behind the numbers their "10" digit in brackets:

5! = 120 (2)
6! = 720 (2)
7! = 5040 (4)
8! = ...20 (2)
9! = ...80 (8)
33 (3)

Let us add the numbers in the brackets:
2+2+4+2+8+3 = 21

Thus, for you sum of factorials the "10" digit is:
1

 Quote by HallsofIvy I presume you meant to say that 10! has last two digits 00
Hi HallsofIvy. I think you've misread courtrigrad's post. It (correctly) says that the sum to 100 has last two digits 13.

Best wishes

X = 7
 I know this sounds dumb, but Im sorry, I dont see the pattern...

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 Quote by chaoseverlasting I know this sounds dumb, but Im sorry, I dont see the pattern...
Did you calculate them out, or are you just looking at the above? I suggest you calculate the (last two digits of) the factorials from 1 to 10, then find the sums of those. The pattern should be obvious.
 Sorry guys... I reworked it and found out I made a mistake... The answer was 71 (taking the last 2 digits of the sum).
 It's natural that the last few digits of of the sum of the factorials 1!+2!+3!+..should be the same. The succeeding terms of the series have all zeros (the number of zeroes depending on how many times 2 and 5 appear in the prime factorization of n!) and so do not affect the first digits of the sums representation.
 1!+2!+3!......50!= wats the answer guys help me out.....
 31035053229546199656252032972759319953190362094566672920420940313
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 Quote by dimitri151 It's natural that the last few digits of of the sum of the factorials 1!+2!+3!+..should be the same. The succeeding terms of the series have all zeros (the number of zeroes depending on how many times 2 and 5 appear in the prime factorization of n!) and so do not affect the first digits of the sums representation.
necroposted!
 Recognitions: Homework Help At least he can use the search button! So they kind of cancel each other out in a way.

 Quote by CRGreathouse necroposted!
 Quote by Mentallic At least he can use the search button! So they kind of cancel each other out in a way.
What grade did you guys say you were in?

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 Quote by dimitri151 What grade did you guys say you were in?
We didn't
 How to find out number of digits in 1!+2!+3!........+100!?