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Number N = 26 × 55 × 76 × 107 how many factors of N are even numbers?
Correct answer is '1183'. Can you explain this answer?
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Number N = 26 × 55 × 76 × 107 how many factors of N are even numbers?...
Factors of N:
To find the factors of N, we need to consider all the possible combinations of multiplying the prime factors of N.
Given that N = 26 × 55 × 76 × 107, we can break it down into prime factors:
N = (2^1 × 13^1) × (5^1 × 11^1) × (2^2 × 19^1) × 107
Finding the Factors:
To find the factors of N, we need to consider all the possible combinations of multiplying the prime factors.
For example, let's take the prime factor 2. We can have 2^0, 2^1, 2^2, and so on. Similarly, for the other prime factors, we can have various powers.
To find the total number of factors, we multiply the number of options for each prime factor together. In this case, we have:
- 2 can have 2 options: 2^0 and 2^1
- 13 has 2 options: 13^0 and 13^1
- 5 has 2 options: 5^0 and 5^1
- 11 has 2 options: 11^0 and 11^1
- 19 has 2 options: 19^0 and 19^1
- 107 has 2 options: 107^0 and 107^1
Multiplying these options together, we get: 2 × 2 × 2 × 2 × 2 × 2 = 2^6 = 64.
Understanding Even Factors:
To determine the number of even factors, we need to consider the power of 2 in each factor. An even factor can be formed by taking any combination of factors where the power of 2 is at least 1.
For example, one such even factor is 2^1 × 13^0 × 5^1 × 11^0 × 19^0 × 107^0.
Calculating the Number of Even Factors:
To calculate the number of even factors, we need to consider the total number of factors and subtract the number of odd factors.
Total number of factors = 64 (from the previous step)
Number of odd factors = 2^0 × 13^0 × 5^0 × 11^0 × 19^0 × 107^0 = 1
Number of even factors = Total number of factors - Number of odd factors
Number of even factors = 64 - 1 = 63
Conclusion:
The correct answer to the question is 63, not 1183 as mentioned.
To find the factors of N, we need to consider all the possible combinations of multiplying the prime factors of N.
Given that N = 26 × 55 × 76 × 107, we can break it down into prime factors:
N = (2^1 × 13^1) × (5^1 × 11^1) × (2^2 × 19^1) × 107
Finding the Factors:
To find the factors of N, we need to consider all the possible combinations of multiplying the prime factors.
For example, let's take the prime factor 2. We can have 2^0, 2^1, 2^2, and so on. Similarly, for the other prime factors, we can have various powers.
To find the total number of factors, we multiply the number of options for each prime factor together. In this case, we have:
- 2 can have 2 options: 2^0 and 2^1
- 13 has 2 options: 13^0 and 13^1
- 5 has 2 options: 5^0 and 5^1
- 11 has 2 options: 11^0 and 11^1
- 19 has 2 options: 19^0 and 19^1
- 107 has 2 options: 107^0 and 107^1
Multiplying these options together, we get: 2 × 2 × 2 × 2 × 2 × 2 = 2^6 = 64.
Understanding Even Factors:
To determine the number of even factors, we need to consider the power of 2 in each factor. An even factor can be formed by taking any combination of factors where the power of 2 is at least 1.
For example, one such even factor is 2^1 × 13^0 × 5^1 × 11^0 × 19^0 × 107^0.
Calculating the Number of Even Factors:
To calculate the number of even factors, we need to consider the total number of factors and subtract the number of odd factors.
Total number of factors = 64 (from the previous step)
Number of odd factors = 2^0 × 13^0 × 5^0 × 11^0 × 19^0 × 107^0 = 1
Number of even factors = Total number of factors - Number of odd factors
Number of even factors = 64 - 1 = 63
Conclusion:
The correct answer to the question is 63, not 1183 as mentioned.
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Community Answer
Number N = 26 × 55 × 76 × 107 how many factors of N are even numbers?...
The prime factorization of 26 × 55 × 76 × 107 is 213 × 512 × 76The total number of factors of N = 14 × 13 × 7 We need to find the total number of even factors. For this, let us find the total number of odd factors and then subtract this from the total number of factors. Any odd factor will have to be a combination of powers of only 5 and 7
Total number of odd factors of 213 × 512 × 76 = (12 + 1) × (6 + 1) = 13 × 7
Total number of factors = (13 + 1) × (12 + 1) × (6 + 1)
Total number of even factors =14 × 13 × 7 − 13 × 7
Number of even factors =13 × 13 × 7 = 1183
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Number N = 26 × 55 × 76 × 107 how many factors of N are even numbers?Correct answer is '1183'. Can you explain this answer?
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Number N = 26 × 55 × 76 × 107 how many factors of N are even numbers?Correct answer is '1183'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Number N = 26 × 55 × 76 × 107 how many factors of N are even numbers?Correct answer is '1183'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Number N = 26 × 55 × 76 × 107 how many factors of N are even numbers?Correct answer is '1183'. Can you explain this answer?.
Number N = 26 × 55 × 76 × 107 how many factors of N are even numbers?Correct answer is '1183'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Number N = 26 × 55 × 76 × 107 how many factors of N are even numbers?Correct answer is '1183'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Number N = 26 × 55 × 76 × 107 how many factors of N are even numbers?Correct answer is '1183'. Can you explain this answer?.
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