Answer
26
Work Step by Step
we need to count the number of factors of 10 in the prime factorization of 100!. Since 10 can be factored as 2 * 5, we need to count the pairs of 2 and 5 in the prime factorization
The number of factors of 2 in the prime factorization of 100! will be greater than or equal to the number of factors of 5. Therefore, we need to count the number of factors of 5
The multiples of 5 in the range 1 to 100 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, and 100. So, there are 20 multiples of 5
However, some of these multiples contribute more than one factor of 5. For example, 25 contributes two factors of 5 (5 * 5), and 50 contributes two factors of 5 (5 * 10). Therefore, we need to consider these multiples as well.
The multiples of 25 in the range 1 to 100 are: 25, 50, 75. So, there are 3 multiples of 25
Since 25 contributes two factors of 5, we count an additional 2 factors of 5 for each multiple of 25
Number of factors of 5 = 20 + 3*2 = 20 + 6 = 26
Therefore, there are 26 zeros at the end of 100!
