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How many factors of 24 x 35 x 104 are perfect squares which are greater than 1?
Correct answer is '44'. Can you explain this answer?
Most Upvoted Answer
How many factors of 24 x 35 x 104 are perfect squares which are great...
24 x 35 x 104
=24 x 35 x 24 * 54
=28 x 35 x 54
For the factor to be a perfect square, the factor should be even power of the number.
In 28, the factors which are perfect squares are 20, 22, 24, 26, 28 = 5
Similarly, in 35, the factors which are perfect squares are 30, 32,34 = 3
In 54, the factors which are perfect squares are 50, 52, 54 = 3
Number of perfect squares greater than 1 = 5*3*3-1
=44
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How many factors of 24 x 35 x 104 are perfect squares which are great...
Solution:
Prime factorization of 24 = 2^3 x 3^1
Prime factorization of 35 = 5^1 x 7^1
Prime factorization of 104 = 2^3 x 13^1
Multiplying these three numbers together gives us:
24 x 35 x 104 = 2^6 x 3^1 x 5^1 x 7^1 x 13^1
Perfect Squares
For a number to be a perfect square, all of its prime factors must have even exponents. Therefore, the prime factorization of a perfect square must be of the form:
p1^e1 x p2^e2 x ... x pn^en, where all the exponents ei are even.
To count the number of perfect squares that are factors of 24 x 35 x 104, we will count the number of possible combinations of even exponents for each prime factor.
Prime factor 2
For the factor to be a perfect square, the exponent of 2 must be even. So, we can choose the exponent of 2 to be 0, 2, 4, or 6.
- 2^0 can be paired with all possible combinations of the other prime factors, giving us 4 possibilities.
- 2^2 can also be paired with all possible combinations of the other prime factors, giving us another 4 possibilities.
- 2^4 can be paired with all possible combinations of the other prime factors except 13^1, giving us 3 possibilities.
- 2^6 can only be paired with 3^1, 5^1, and 7^1, giving us 3 possibilities.
Total number of possibilities for prime factor 2 = 4 + 4 + 3 + 3 = 14
Prime factor 3
For the factor to be a perfect square, the exponent of 3 must be even. So, we can choose the exponent of 3 to be 0 or 2.
- 3^0 can be paired with all possible combinations of the other prime factors, giving us 4 possibilities.
- 3^2 can only be paired with 2^0, 5^1, and 7^1, giving us 3 possibilities.
Total number of possibilities for prime factor 3 = 4 + 3 = 7
Prime factor 5
For the factor to be a perfect square, the exponent of 5 must be even. So, we can choose the exponent of 5 to be 0 or 2.
- 5^0 can be paired with all possible combinations of the other prime factors, giving us 4 possibilities.
- 5^2 can only be paired with 2^0 and 3^0, giving us 2 possibilities.
Total number of possibilities for prime factor 5 = 4 + 2 = 6
Prime factor 7
For the factor to be a perfect square, the exponent of 7 must be even. So, we can choose the exponent of 7 to be 0 or 2.
- 7^0 can be paired with all possible combinations of the other prime factors, giving us 4 possibilities.
- 7^2 can only be paired with 2^0 and 3^0, giving us 2 possibilities.
Prime factorization of 24 = 2^3 x 3^1
Prime factorization of 35 = 5^1 x 7^1
Prime factorization of 104 = 2^3 x 13^1
Multiplying these three numbers together gives us:
24 x 35 x 104 = 2^6 x 3^1 x 5^1 x 7^1 x 13^1
Perfect Squares
For a number to be a perfect square, all of its prime factors must have even exponents. Therefore, the prime factorization of a perfect square must be of the form:
p1^e1 x p2^e2 x ... x pn^en, where all the exponents ei are even.
To count the number of perfect squares that are factors of 24 x 35 x 104, we will count the number of possible combinations of even exponents for each prime factor.
Prime factor 2
For the factor to be a perfect square, the exponent of 2 must be even. So, we can choose the exponent of 2 to be 0, 2, 4, or 6.
- 2^0 can be paired with all possible combinations of the other prime factors, giving us 4 possibilities.
- 2^2 can also be paired with all possible combinations of the other prime factors, giving us another 4 possibilities.
- 2^4 can be paired with all possible combinations of the other prime factors except 13^1, giving us 3 possibilities.
- 2^6 can only be paired with 3^1, 5^1, and 7^1, giving us 3 possibilities.
Total number of possibilities for prime factor 2 = 4 + 4 + 3 + 3 = 14
Prime factor 3
For the factor to be a perfect square, the exponent of 3 must be even. So, we can choose the exponent of 3 to be 0 or 2.
- 3^0 can be paired with all possible combinations of the other prime factors, giving us 4 possibilities.
- 3^2 can only be paired with 2^0, 5^1, and 7^1, giving us 3 possibilities.
Total number of possibilities for prime factor 3 = 4 + 3 = 7
Prime factor 5
For the factor to be a perfect square, the exponent of 5 must be even. So, we can choose the exponent of 5 to be 0 or 2.
- 5^0 can be paired with all possible combinations of the other prime factors, giving us 4 possibilities.
- 5^2 can only be paired with 2^0 and 3^0, giving us 2 possibilities.
Total number of possibilities for prime factor 5 = 4 + 2 = 6
Prime factor 7
For the factor to be a perfect square, the exponent of 7 must be even. So, we can choose the exponent of 7 to be 0 or 2.
- 7^0 can be paired with all possible combinations of the other prime factors, giving us 4 possibilities.
- 7^2 can only be paired with 2^0 and 3^0, giving us 2 possibilities.
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How many factors of 24 x 35 x 104 are perfect squares which are greater than 1?Correct answer is '44'. Can you explain this answer?
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How many factors of 24 x 35 x 104 are perfect squares which are greater than 1?Correct answer is '44'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about How many factors of 24 x 35 x 104 are perfect squares which are greater than 1?Correct answer is '44'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many factors of 24 x 35 x 104 are perfect squares which are greater than 1?Correct answer is '44'. Can you explain this answer?.
How many factors of 24 x 35 x 104 are perfect squares which are greater than 1?Correct answer is '44'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about How many factors of 24 x 35 x 104 are perfect squares which are greater than 1?Correct answer is '44'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many factors of 24 x 35 x 104 are perfect squares which are greater than 1?Correct answer is '44'. Can you explain this answer?.
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