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 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? 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 Hello! 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 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
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