Number of Zeroes in 60!

To determine the number of zeroes in 60!, we need to find the number of factors of 10 in the prime factorization of 60!. Since 10 is the product of 2 and 5, we need to find the number of pairs of 2 and 5 in the prime factorization of 60!.

Prime factorization of 60
To find the prime factorization of 60, we divide it by prime numbers starting from 2 until we cannot divide it any further. The prime factorization of 60 is as follows:

60 = 2 * 30
= 2 * 2 * 15
= 2 * 2 * 3 * 5

Counting Factors of 2 and 5
From the prime factorization of 60, we can see that there are two factors of 2 and one factor of 5. However, we need to consider that 2 is a factor of 4, 8, 16, etc., and 5 is a factor of 25, 50, etc. So, we need to count the number of multiples of 2 and 5 in the range from 1 to 60.

Multiples of 2:
To count the factors of 2, we divide 60 by increasing powers of 2 until the quotient is less than 2. We get:

60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 2 = 7.5 (not a whole number)

So, there are 30 + 15 + 7 = 52 factors of 2 in the range from 1 to 60.

Multiples of 5:
To count the factors of 5, we divide 60 by increasing powers of 5 until the quotient is less than 5. We get:

60 ÷ 5 = 12
12 ÷ 5 = 2.4 (not a whole number)

So, there are 12 + 2 = 14 factors of 5 in the range from 1 to 60.

Counting Pairs of 2 and 5
Since there are 52 factors of 2 and 14 factors of 5, we can only form 14 pairs of 2 and 5. Each pair contributes a trailing zero to the factorial. Therefore, there are 14 zeroes in the number 60!