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Problem Set 1 Question 10 How many zeros are at the end of the number 100!? Solution 1 We can ask Mathematica to compute the number 100! and count the number of zeros at the end. After entering the command 100! Mathematica returned the value 9332621544394415268169923885626670049071596826438162146859296389521 7599993229915608941463976156518286253697920827223758251185210916864 000000000000000000000000 Now we just count the zeros at the end. We see there are 24 zeros at the end of 100! Solution 2 To get a zero at the end of a number we need to count the number of pairs of 2 and 5 that are in the prime factorization of the number. In any factorial there will always be more factors of 2 than factors of 5 it suffices to count the factors of 5 since there will be plenty of 2's to pair with them. We break 100! into factors and count factors of 5. 1 2 3 4 5-1 6 7 8 9 10 - 1 2 11 12 13 14 15 - 1 16 17 18 19 20 - 1 21 22 23 24 25 - 2 26 27 28 29 30 - 1 31 32 33 34 35 - 1 36 37 38 39 40 - 1 41 42 43 44 45 - 1 46 47 48 49 50 - 2 51 52 53 54 55 - 1 56 57 58 59 60 - 1 61 62 63 64 65 - 1 66 67 68 69 70 - 1 71 72 73 74 75 - 2 76 77 78 79 80 - 1 81 82 83 84 85 - 1 86 87 88 89 90 - 1 91 92 93 94 95 - 1 96 97 98 99 100 - 2 2 3 2 3 2 2 3 2 3 Add up the number of factors of 5: 2 + 2 + 3 + 2 + 3 + 2 + 2 + 3 + 2 + 3 = 24 We see there are 24 factors of 5 so in conclusion the number of zeros at the end of 100! is going to be 24.