Date: Thu, 20 Jul 2000 20:53:01 EDT Hi I was just wondering if you could tell me how many zeros are in 100,000! (factorial.) I am a college sophomore but i believe this could be considered a highschool level question. I would appreciate your help. Thank you.
sincerely Let's try 25! instead. This is the general idea. What power of 10 divides 25!? 10 = 2x5 thus every time 10 divides 25! we need a 2 and a 5 to divide 25! How many 5 divide 25!? 5 divides every 5th number (5, 10, 15, 20, 25) and 5^{2} divides 25 thus there are 6 5's in 25! How many 2's? 2 divides every second number, 2^{2} divides every 4th number, 2^{3} divides every 8th number and 2^{4} divides every 16th number, thus the number of 2 that appear is much larger than the number of 5's. Since there were only 25/5 + 25/5^{2} = 5 + 1 = 6 5s in 25! we only get 10 to divide 25! 6 times in spite of the presence of all those 2s. For 200!, 10 divides it 200/5 + 200/5^{2} + 200/5^{3} = 40 + 8 + 1 = 49 times. (Notice here that 200/5^{3} = 200/125 = 1.6 but we want only the integer part, 1.) For 100,000 the argument is the same.
etc. In total 100,000/5 + 100,000/5^{2} + 100,000/5^{3} + ... times. (Again in each division you want only the integer part.) Cheers,Denis
