The Number of Zeros in 60 Factorial
Factorials are mathematical operations commonly encountered in various fields of study, including number theory and combinatorics. A factorial is denoted by an exclamation mark (!) and represents the product of all positive integers from 1 to a given number. For instance, 5! (read as “5 factorial”) is equal to 5 × 4 × 3 × 2 × 1, which equals 120.
One interesting question that arises when dealing with factorials is determining the number of zeros at the end of a factorial. In this article, we will focus on finding the number of zeros in 60 factorial (60!). We will explore this topic based on information obtained from reputable sources, including the 2IIM CAT Questions and Answers platform, Vedantu, and CoolConversion.
Understanding the Concept
To determine the number of zeros at the end of a factorial, we need to analyze the prime factorization of the factorial number. The crucial insight lies in recognizing that a zero is formed by multiplying 2 and 5. Consequently, finding the number of pairs of 2 and 5 in the prime factorization will give us the number of zeros at the end of the factorial.
It is important to note that the number of 2s in the prime factorization of a factorial is always greater than or equal to the number of 5s. Therefore, we focus on finding the number of 5s in the prime factorization to determine the number of zeros.
Determining the Number of Zeros
To find the number of zeros at the end of 60 factorial, we need to count the number of 5s in its prime factorization. Let’s break down the process step by step:
- Prime Factorization of 60: The prime factorization of 60 is 2^2 × 3 × 5.
- Counting the Number of 5s: In the prime factorization of 60, there is only one 5. Hence, there is one pair of 2 and 5, resulting in one zero at the end of 60!.
Conclusion
In conclusion, the number of zeros at the end of 60 factorial (60!) is one. This result is derived by analyzing the prime factorization of 60 and counting the number of 5s, which correspond to the number of pairs of 2 and 5. The concept of finding trailing zeros in factorials is essential in number theory and combinatorics.
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FAQs
What is the concept of counting zeros in a factorial?
The concept of counting zeros in a factorial involves determining the number of zeros at the end of the factorial number. This is done by analyzing the prime factorization and identifying the number of pairs of 2 and 5.
How are zeros formed in a factorial?
Zeros are formed in a factorial by multiplying pairs of 2 and 5. Since 10 is equal to 2 × 5, each pair of 2 and 5 contributes one zero at the end of the factorial.
Why do we focus on the number of 5s in the prime factorization?
In the prime factorization of a factorial, the number of 2s is always greater than or equal to the number of 5s. Therefore, by counting the number of 5s, we can determine the number of pairs of 2 and 5 and, consequently, the number of zeros at the end of the factorial.
How can we determine the number of zeros in 60 factorial?
To determine the number of zeros in 60 factorial (60!), we need to examine its prime factorization. Count the number of 5s in the prime factorization of 60 and that will give you the number of zeros at the end of 60!.
What is the prime factorization of 60?
The prime factorization of 60 is 2^2 × 3 × 5. It consists of two 2s, one 3, and one 5.
How many 5s are there in the prime factorization of 60?
In the prime factorization of 60, there is only one 5.
How does the number of 5s relate to the number of zeros in 60 factorial?
The number of 5s in the prime factorization of 60 factorial corresponds to the number of pairs of 2 and 5. Each pair contributes one zero at the end of the factorial.
What is the final count of zeros in 60 factorial?
The prime factorization of 60 factorial contains one pair of 2 and 5, resulting in one zero at the end of 60 factorial.