Number of Trailing Zeros in 60 Factorial

Factorials play a significant role in various mathematical calculations, and determining the number of trailing zeros in a factorial is an important aspect of number theory. This article aims to explore the concept of trailing zeros and provide a detailed explanation of how to determine the number of trailing zeros in 60 factorial, drawing upon information from the provided resources.

Definition and Explanation of Factorial

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers from 1 to n. For instance, 5! = 5 * 4 * 3 * 2 * 1 = 120. Factorials are commonly encountered in combinatorics, probability, and other mathematical applications.

Trailing Zeros in Factorials

Trailing zeros refer to the consecutive zeros at the end of a number. In the context of factorials, trailing zeros are of particular interest because they provide insights into the factors of 10 present in the factorial. The presence of trailing zeros is directly related to the factors of 5 and 2 in the factorial, as 10 is the product of 2 and 5.

Method for Counting Trailing Zeros

To determine the number of trailing zeros in a factorial, we can follow a systematic approach:

1. Identify the highest power of 5 that divides the factorial. This can be done by repeatedly dividing the factorial by 5 until the result is no longer divisible by 5.
2. Identify the highest power of 2 that divides the factorial. This can be done by repeatedly dividing the factorial by 2 until the result is no longer divisible by 2.
3. The number of trailing zeros is the minimum of the powers of 5 and 2 obtained in steps 1 and 2.

Applying the Method to 60 Factorial

Let’s apply the aforementioned method to determine the number of trailing zeros in 60 factorial:

1. Highest power of 5:

   60! / 5 = 12!

   12! / 5 = 2!

   2! / 5 = 0

Therefore, the highest power of 5 that divides 60! is 12.

2. Highest power of 2:

   60! / 2 = 30!

   30! / 2 = 15!

   15! / 2 = 7!

   7! / 2 = 3!

   3! / 2 = 1!

   1! / 2 = 0

Therefore, the highest power of 2 that divides 60! is 28.

3. Number of trailing zeros:

   The minimum of 12 and 28 is 12.

Hence, there are 12 trailing zeros in 60 factorial.

Additional Examples

To further illustrate the concept, let’s consider a few more examples:

1. 10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

• Highest power of 5: 10! / 5 = 2!
• Highest power of 2: 10! / 2 = 5!
• Number of trailing zeros: Minimum of 2 and 5 is 2.

2. 25! = 25 * 24 * 23 * … * 3 * 2 * 1

• Highest power of 5: 25! / 5 = 5!
• Highest power of 2: 25! / 2 = 12!
• Number of trailing zeros: Minimum of 5 and 12 is 5.

Summary and Conclusion

In summary, the number of trailing zeros in a factorial is determined by identifying the highest powers of 5 and 2 that divide the factorial and taking the minimum of these powers. Applying this method to 60 factorial, we found that there are 12 trailing zeros. The concept of trailing zeros is significant in various mathematical applications, and understanding this concept is essential for solving problems related to factorials and combinatorics.

References:

1. https://www.vedantu.com/question-answer/the-number-of-zeros-at-the-end-of-60-is-class-11-maths-cbse-5fec734241231c3a78698ea9
2. https://www.toppr.com/ask/question/the-number-of-zeros-at-the-end-of-60-is-2/
3. https://iim-cat-questions-answers.2iim.com/quant/number-system/factorial/factorial_5.shtml

FAQs

What are trailing zeros?

Trailing zeros are the consecutive zeros at the end of a number. In the context of factorials, trailing zeros provide insights into the factors of 10 present in the factorial.

Why are trailing zeros significant in factorials?

Trailing zeros in factorials are significant because they are related to the factors of 10 present in the factorial. The number of trailing zeros indicates the highest power of 10 that divides the factorial.

How do you determine the number of trailing zeros in a factorial?

To determine the number of trailing zeros in a factorial, you can follow these steps:

1. Identify the highest power of 5 that divides the factorial.
2. Identify the highest power of 2 that divides the factorial.
3. The number of trailing zeros is the minimum of the powers of 5 and 2 obtained in steps 1 and 2.

Can you provide an example of calculating trailing zeros in a factorial?

Sure. Let’s calculate the number of trailing zeros in 60 factorial:

1. Highest power of 5: 60! / 5 = 12!
2. Highest power of 2: 60! / 2 = 30!
3. Number of trailing zeros: Minimum of 12 and 28 is 12.

   Therefore, there are 12 trailing zeros in 60 factorial.

Are there any patterns or observations related to trailing zeros in factorials?

Yes, there are a few patterns and observations related to trailing zeros in factorials:

• The number of trailing zeros is always related to the factors of 5 and 2 present in the factorial.
• For a factorial to have trailing zeros, it must contain at least one factor of 5 and one factor of 2.
• The number of trailing zeros increases as the factorial number increases.