Associated Topics || Dr. Math Home || Search Dr. Math

### Find the Last Eight Digits of a Factorial

```
Date: 1/30/96 at 7:3:23
From: Bipin D Patel
Subject: A Quick question

Hello Dr. Maths,

Please pardon me for bothering you with this question.  I have
been trying to access the information I require from your library
services but I keep getting lost in the huge system. I saw a list
of people associated with the library services and thought it
might be simpler to ask someone.   My apologies for taking up your
time.  I have two questions that I am trying to find the
information on through the Internet and I have only managed to get
some of the information. Perhaps you might be able to help or
point me in the right direction.

Q1. Find the last eight digits of 69! where 69!=69x68x67........x1

if a = 69x68x67x66x65x64x63x62x61    = 2.056516253536e+16
if b = 60x59x58x57x56x55x54x53x52x51 = 2.735898472315e+17
if c = 50x49x48x47x46x45x44x43x42x41 = 3.72760430233e+16
if d = 40x39x38x37x36x35x34x33x32x31 = 3.075990524006e+15
if e = 30x29x28x27x26x25x24x23x22x21 = 1.09027350432e+14
if f = 20x19x18x17x16x15x14x13x12x11 = 670442572800
if g = 10x9x8x7x6x5x4x3x2x1          = 3628800

therefore h=axbxcxdxexfxg
h=1711.224524282e+95
According to my recollection of basic maths any number ending with
a zero multiplied by any other number always ends as a number with
a zero.
i.e 30 x 3 =90

So would I be right in assuming that the last eight digits of
69!=h=1711224524282.....00000000

Or should I give up maths altogether!

Q2. What does the V stand for in V1 & V2 rockets?

I think the V rockets were designed and built by Dr Werner Von
Braun and the V stood for "Vengeance" weapon. Would this be
correct? what is the German word be for Vengeance?

Bipin D Patel
Processing Centre Manager
AT&T Global Information Solutions
```

```
Date: 2/8/96 at 22:11:53
From: Doctor Ken
Subject: Re: A Quick question

Hello!

8 digits of 69! are indeed zeroes.  You should keep in mind that
the number 2.056516253536e+16 is only an approximation to
69x68x67x66x65x64x63x62x61, though.  The real number is
20565162535357440.

The easy way to see that there are a bunch of zeroes at the end of
69! is to notice that every 0 at the end of a number means that 10
is a factor of the number, which means that 2 and 5 must be
factors.  So to find out how many 0's are at the end, you just
need to find out how many pairs of 2's and 5's there are in 69!.
And since there will be more 2's than 5's (do you know why?) you
really only need to count the 5's.  The 5's will come from these
factors of 69!: 5,10,15,20,25,30,35,40,45,50,55,60,65 - that's 13
numbers, and actually 25 and 50 each have 2 5's in them, so you'll
have 15 5's in 69!, and thus 15 zeroes.

As for the rocket question, I'm afraid I have no idea.  Too bad
there's not a Doctor Rocketry!

-Doctor Ken,  The Math Forum

```
Associated Topics:
Middle School Factorials

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search