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What will be the number of zero s at the end of product of first 100 natural numbers?
Most Upvoted Answer
What will be the number of zero s at the end of product of first 100 n...
To get 0s, we have to make 2×5 pairs
now since 2 will be more than 5, so we just have to count the number of 5
in 100 ! there are 24 5s ( 100 / 20 and 20 /5 ie 20 + 4 ) by the method of factorization
so total 24 zeros
now since 2 will be more than 5, so we just have to count the number of 5
in 100 ! there are 24 5s ( 100 / 20 and 20 /5 ie 20 + 4 ) by the method of factorization
so total 24 zeros
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What will be the number of zero s at the end of product of first 100 n...
Number of Zeros at the end of the Product of First 100 Natural Numbers
To determine the number of zeros at the end of the product of the first 100 natural numbers, we need to analyze the factors that contribute to the creation of zeros.
Prime Factors of 10
To understand the creation of zeros, we need to consider the prime factors of 10. The prime factors of 10 are 2 and 5.
Multiplication of 2 and 5
A zero is created when a number is multiplied by 10, which is the product of 2 and 5. Therefore, to create a zero at the end of a number, it must be multiplied by both 2 and 5.
Frequency of 2 and 5
In the first 100 natural numbers, there are many more multiples of 2 than 5. This is because every even number is divisible by 2, while only multiples of 5 (5, 10, 15, 20, etc.) contain the factor 5.
Counting the Frequency of 5
To determine the number of zeros at the end of the product, we need to count the frequency of 5 in the first 100 natural numbers.
Counting Multiples of 5
There are 100 natural numbers, and every multiple of 5 is counted only once. Therefore, the number of multiples of 5 in the first 100 natural numbers can be calculated using the formula:
Number of multiples of 5 = floor(100/5) = 20
This means there are 20 numbers in the range of 1 to 100 that are divisible by 5.
Counting Powers of 5
There might be numbers that have multiple factors of 5, such as 25, 50, and 75. To account for these numbers, we need to count the powers of 5.
Number of multiples of 25 = floor(100/25) = 4
Number of multiples of 125 = floor(100/125) = 0 (as there are no multiples of 125 in the range of 1 to 100)
Counting the Total Frequency of 5
To determine the total frequency of 5 in the first 100 natural numbers, we can add up the counts of multiples and powers of 5:
Total frequency of 5 = Number of multiples of 5 + Number of multiples of 25 + Number of multiples of 125 = 20 + 4 + 0 = 24
Therefore, there are a total of 24 factors of 5 in the product of the first 100 natural numbers.
Number of Zeros
Since every zero is created by multiplying a number by 10 (which is 2 * 5), we need to ensure that we have an equal number of 2s and 5s. In this case, we have 24 factors of 5, but there are more than enough 2s available in the first 100 natural numbers to pair with these factors.
Therefore, the number of zeros at the end of the product of the first 100 natural numbers will be equal to the number of factors of 5, which is 24.
To determine the number of zeros at the end of the product of the first 100 natural numbers, we need to analyze the factors that contribute to the creation of zeros.
Prime Factors of 10
To understand the creation of zeros, we need to consider the prime factors of 10. The prime factors of 10 are 2 and 5.
Multiplication of 2 and 5
A zero is created when a number is multiplied by 10, which is the product of 2 and 5. Therefore, to create a zero at the end of a number, it must be multiplied by both 2 and 5.
Frequency of 2 and 5
In the first 100 natural numbers, there are many more multiples of 2 than 5. This is because every even number is divisible by 2, while only multiples of 5 (5, 10, 15, 20, etc.) contain the factor 5.
Counting the Frequency of 5
To determine the number of zeros at the end of the product, we need to count the frequency of 5 in the first 100 natural numbers.
Counting Multiples of 5
There are 100 natural numbers, and every multiple of 5 is counted only once. Therefore, the number of multiples of 5 in the first 100 natural numbers can be calculated using the formula:
Number of multiples of 5 = floor(100/5) = 20
This means there are 20 numbers in the range of 1 to 100 that are divisible by 5.
Counting Powers of 5
There might be numbers that have multiple factors of 5, such as 25, 50, and 75. To account for these numbers, we need to count the powers of 5.
Number of multiples of 25 = floor(100/25) = 4
Number of multiples of 125 = floor(100/125) = 0 (as there are no multiples of 125 in the range of 1 to 100)
Counting the Total Frequency of 5
To determine the total frequency of 5 in the first 100 natural numbers, we can add up the counts of multiples and powers of 5:
Total frequency of 5 = Number of multiples of 5 + Number of multiples of 25 + Number of multiples of 125 = 20 + 4 + 0 = 24
Therefore, there are a total of 24 factors of 5 in the product of the first 100 natural numbers.
Number of Zeros
Since every zero is created by multiplying a number by 10 (which is 2 * 5), we need to ensure that we have an equal number of 2s and 5s. In this case, we have 24 factors of 5, but there are more than enough 2s available in the first 100 natural numbers to pair with these factors.
Therefore, the number of zeros at the end of the product of the first 100 natural numbers will be equal to the number of factors of 5, which is 24.
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What will be the number of zero s at the end of product of first 100 natural numbers?
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What will be the number of zero s at the end of product of first 100 natural numbers? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about What will be the number of zero s at the end of product of first 100 natural numbers? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What will be the number of zero s at the end of product of first 100 natural numbers?.
What will be the number of zero s at the end of product of first 100 natural numbers? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about What will be the number of zero s at the end of product of first 100 natural numbers? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What will be the number of zero s at the end of product of first 100 natural numbers?.
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