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The integers r and s are multiples of 12 and 75, respectively. What is the minimum number of factors of r × s?
Correct answer is '27'. Can you explain this answer?
Most Upvoted Answer
The integers r and s are multiples of 12 and 75, respectively. What is...
Since r is a multiple of 12, we have:
r = 12 × m, where m is a positive integer
= 22 × 3 × m
Again, since s is a multiple of 75, we have:
Again, since s is a multiple of 75, we have:
s = 75 × n, where n is a positive integer
= 3 × 52 × n
⇒ r × s = 22 × 32 × 52 × m × n
To minimize the number of factors of r×s, we need to take m = n = 1
⇒ r × s = 22 × 32 × 52
We know that for αm × bn × cp, the number of factors = (m+1)(n+1)(p+1); where α, b, and c are prime factors
Thus, the minimum number of factors = (2+1)(2+1)(2+1) = 3 × 3 × 3 = 27
The correct answer is 27.
⇒ r × s = 22 × 32 × 52 × m × n
To minimize the number of factors of r×s, we need to take m = n = 1
⇒ r × s = 22 × 32 × 52
We know that for αm × bn × cp, the number of factors = (m+1)(n+1)(p+1); where α, b, and c are prime factors
Thus, the minimum number of factors = (2+1)(2+1)(2+1) = 3 × 3 × 3 = 27
The correct answer is 27.
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The integers r and s are multiples of 12 and 75, respectively. What is...
In order to find the minimum number of factors of r, we need to find the prime factorization of r.
Since r is a multiple of 12, we know that it can be expressed as r = 12 * x, where x is an integer.
Since r is also a multiple of 75, we know that it can be expressed as r = 75 * y, where y is an integer.
Combining these two expressions, we have 12 * x = 75 * y.
To find the prime factorization of r, we need to find the common factors of 12 and 75.
The prime factorization of 12 is 2^2 * 3, and the prime factorization of 75 is 3 * 5^2.
The common factors of 12 and 75 are 3 and 3.
Therefore, the prime factorization of r is 2^2 * 3 * 3 = 12 * 3 = 36.
To find the number of factors of 36, we need to consider the exponents of each prime factor in the prime factorization.
The exponent for 2 is 2, so we have 2^0, 2^1, and 2^2 as possible factors.
The exponent for 3 is 2, so we have 3^0, 3^1, and 3^2 as possible factors.
Therefore, the minimum number of factors of r is (2+1) * (2+1) = 3 * 3 = 9.
Since r is a multiple of 12, we know that it can be expressed as r = 12 * x, where x is an integer.
Since r is also a multiple of 75, we know that it can be expressed as r = 75 * y, where y is an integer.
Combining these two expressions, we have 12 * x = 75 * y.
To find the prime factorization of r, we need to find the common factors of 12 and 75.
The prime factorization of 12 is 2^2 * 3, and the prime factorization of 75 is 3 * 5^2.
The common factors of 12 and 75 are 3 and 3.
Therefore, the prime factorization of r is 2^2 * 3 * 3 = 12 * 3 = 36.
To find the number of factors of 36, we need to consider the exponents of each prime factor in the prime factorization.
The exponent for 2 is 2, so we have 2^0, 2^1, and 2^2 as possible factors.
The exponent for 3 is 2, so we have 3^0, 3^1, and 3^2 as possible factors.
Therefore, the minimum number of factors of r is (2+1) * (2+1) = 3 * 3 = 9.
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The integers r and s are multiples of 12 and 75, respectively. What is the minimum number of factors of r × s?Correct answer is '27'. Can you explain this answer? for GRE 2025 is part of GRE preparation. The Question and answers have been prepared according to the GRE exam syllabus. Information about The integers r and s are multiples of 12 and 75, respectively. What is the minimum number of factors of r × s?Correct answer is '27'. Can you explain this answer? covers all topics & solutions for GRE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The integers r and s are multiples of 12 and 75, respectively. What is the minimum number of factors of r × s?Correct answer is '27'. Can you explain this answer?.
The integers r and s are multiples of 12 and 75, respectively. What is the minimum number of factors of r × s?Correct answer is '27'. Can you explain this answer? for GRE 2025 is part of GRE preparation. The Question and answers have been prepared according to the GRE exam syllabus. Information about The integers r and s are multiples of 12 and 75, respectively. What is the minimum number of factors of r × s?Correct answer is '27'. Can you explain this answer? covers all topics & solutions for GRE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The integers r and s are multiples of 12 and 75, respectively. What is the minimum number of factors of r × s?Correct answer is '27'. Can you explain this answer?.
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