# Factorial Sum

1. Dec 1, 2006

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.

2. Dec 1, 2006

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 $$\sum_{n=1}^{100} n!$$

Last edited: Dec 1, 2006
3. Dec 2, 2006

### HallsofIvy

Staff Emeritus
I presume you meant to say that 10! has last two digits 00, not $$\sum_{n=1}^{100} n!$$.

4. Dec 2, 2006

### Edgardo

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

5. Dec 2, 2006

### X=7

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

6. Dec 4, 2006

### chaoseverlasting

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

7. Dec 4, 2006

### CRGreathouse

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.

8. Dec 5, 2006

Sorry guys... I reworked it and found out I made a mistake... The answer was 71 (taking the last 2 digits of the sum).

9. Aug 2, 2010

### 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.

10. Aug 17, 2010

### gowtham_012

1!+2!+3!......50!=
wats the answer guys help me out.....

11. Aug 17, 2010

### dimitri151

31035053229546199656252032972759319953190362094566672920420940313

12. Aug 18, 2010

### Mentallic

lol ripped

13. Aug 18, 2010

### CRGreathouse

necroposted!

14. Aug 18, 2010

### Mentallic

At least he can use the search button! So they kind of cancel each other out in a way.

15. Aug 19, 2010

### dimitri151

What grade did you guys say you were in?

16. Aug 20, 2010

### Mentallic

We didn't :tongue:

17. Mar 10, 2012

### piyushon2411

How to find out number of digits in 1!+2!+3!........+100!?

18. Mar 10, 2012

### chiro

Hey piyushon2411 and welcome to the forums.

For this problem in base b you need to calculate log_b(z) = ln(z)/ln(b) where b is the number of possibilities in each digit. Round up if you have a fractional part in your answer.

The z is your expression 1!+ 2! + 3! + ... blah

What you should realize is that for this problem you only need to evaluate the highest term which is 100!.

Using properties of logs you can use the property log(ab) = log(a) + log(b) which means the answer for this problem is:

log(1) + log(2) + .... log(100) from 1 up to 100 like the sum suggests.

19. Mar 10, 2012

### Mensanator

The answer IS 13. I think you miscounted the 10s. I get 8+2+4+2+2+2

20. Mar 15, 2012

### piyushon2411

Hey chiro, thanks for the solution,
i doubt that i got it fully ,can u please explain wt u did after we consider just 100! to get the number of digits,thanks in advance

For the above problm to get the last 2 digits,
going by the conventnl method, last 2 digits from 1 to 9 factorial goes like
01+02+06+24+20+20+40+20+80= 13 in the last 2 digits