How Many Factors Of 10 In 100! [duplicate]

haidar

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Sep 25, 2024

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The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120. Determining the number of factors of 10 in a factorial involves understanding the concept of prime factorization and how it relates to the formation of factors of 10.

Understanding the Factorial and Factors of 10

To find the number of factors of 10 in 100!, we need to understand how factors of 10 are formed. A factor of 10 is formed by the product of 2 and 5. Therefore, we need to count the number of pairs of 2 and 5 present within the prime factorization of 100!.

Prime Factorization of 100!

The prime factorization of 100! involves identifying all prime numbers that are factors of 100! and their respective powers. To do this, we can use the following logic:

• Counting the Number of 2s: In the sequence 1 to 100, there are 50 even numbers (divisible by 2). Each even number contributes one factor of 2. Additionally, there are 25 numbers divisible by 4, each contributing another factor of 2. Similarly, there are 12 numbers divisible by 8, 6 divisible by 16, 3 divisible by 32, and 1 divisible by 64, contributing additional factors of 2. Therefore, the total number of factors of 2 in 100! is 50 + 25 + 12 + 6 + 3 + 1 = 97.

• Counting the Number of 5s: The process for counting factors of 5 is similar. There are 20 numbers divisible by 5, 4 divisible by 25, and 1 divisible by 100. This gives us a total of 20 + 4 + 1 = 25 factors of 5 in 100!.

Finding the Number of Factors of 10

Since each factor of 10 requires one factor of 2 and one factor of 5, the number of factors of 10 in 100! is determined by the smaller of the two counts for factors of 2 and 5. In this case, we have 25 factors of 5 and 97 factors of 2. Therefore, there are 25 factors of 10 in 100!.

Generalizing the Process

This approach can be generalized to find the number of factors of 10 in any factorial.

1. Determine the prime factorization of the factorial: Identify all the prime numbers that are factors of the factorial and their respective powers.
2. Count the number of factors of 2 and 5: Use the same logic as explained above to count the factors of 2 and 5.
3. The minimum count determines the number of factors of 10: The smaller of the two counts for factors of 2 and 5 represents the number of factors of 10.

Conclusion

The number of factors of 10 in a factorial can be determined by analyzing the prime factorization of the factorial. We found that there are 25 factors of 10 in 100!. Understanding this concept allows us to efficiently calculate the number of factors of 10 for any factorial, which is useful in various mathematical applications.