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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

My answer:

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
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 

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?

Many thanks for your assistance.

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


Well, you're absolutely right about your first question.  The last 
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 

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

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