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If the product of two positive integers is 540, which of the following can be the least common multiple and the greatest
common divisor respectively of the two integers?
I. 108 and 5
II. 90 and 6
III. 27 and 20
  • a)
    I only
  • b)
    II only
  • c)
    III only
  • d)
    I, II and III
  • e)
    None of the above
Correct answer is option 'B'. Can you explain this answer?

Answers

Given:
  • Let the 2 positive integers be A and B
  • A*B = 540
To Find: Which of the 3 pairs of numbers can be LCM(A,B) and GCD(A,B) respectively
Approach:
  1. To answer the question, we’ll evaluate the constraints on LCM(A,B) and GCD(A,B):
    1. Constraint 1: LCM(A,B) * GCD(A,B) = A*B
    2. Constraint 2: The LCM(A,B) contains the highest power EACH prime factor of A and B.
      • This means, every prime factor that occurs in either one or both of A and B is represented in the LCM(A,B)
        1. So, every prime factor that occurs in the product of A and B will also occur in LCM(A,B)
  2. Next, we’ll find which of the 3 pairs satisfy both the above constraints
Working out:
  • Evaluating the 2 constraints
    • Constraint 1: LCM(A,B) * GCD(A,B) = 540
    • Constraint 2:
      • 540 = 22 *33 *5
        • The prime factors of 540 are 2, 3 and 5
      • So, the prime factors of LCM(A,B) are 2, 3 and 5
  • Checking the 3 pairs
    • 108 and 5
      • The product of 108 and 5 is 540. So, the first Constraint is satisfied
      • 108 = 2 *3
        • The prime factors of 108 are not 2, 3 and 5. So, Constraint 2 is not satisfied
        • So, this pair is rejected
  • 90 and 6
    • The product of 90 and 6 is 540. So, the first Constraint is satisfied
    • 90 = 2*3 *5
      • The prime factors of 90 are 2, 3 and 5. So, Constraint 2 is also satisfied
    • So, this pair is possible
 
  • 27 and 20
  • The product of 27 and 20 is 540. So, the first Constraint is satisfied
  • 27 = 33
    • The prime factors of 27 are not 2, 3 and 5. So, Constraint 2 is not satisfied
  • So, this pair is rejected
Looking at the answer choices, we see that the correct answer is Option B

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If the product of two positive integers is 540, which of the following can be the least common multiple and the greatestcommon divisor respectively of the two integers?I. 108 and 5II. 90 and 6III. 27 and 20a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer? for GMAT 2023 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If the product of two positive integers is 540, which of the following can be the least common multiple and the greatestcommon divisor respectively of the two integers?I. 108 and 5II. 90 and 6III. 27 and 20a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2023 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the product of two positive integers is 540, which of the following can be the least common multiple and the greatestcommon divisor respectively of the two integers?I. 108 and 5II. 90 and 6III. 27 and 20a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'B'. Can you explain this answer?.
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Given: Let the 2 positive integers be A and B A*B = 540To Find:Which of the 3 pairs of numbers can be LCM(A,B) and GCD(A,B) respectivelyApproach: To answer the question, well evaluate the constraints on LCM(A,B) and GCD(A,B): Constraint 1: LCM(A,B) * GCD(A,B) = A*B Constraint 2: The LCM(A,B) contains the highest power EACH prime factor of A and B. This means, every prime factor that occurs in either one or both of A and B is represented in the LCM(A,B) So, every prime factor that occurs in the product of A and B will also occur in LCM(A,B) Next, well find which of the 3 pairs satisfy both the above constraintsWorking out: Evaluating the 2 constraints Constraint 1: LCM(A,B) * GCD(A,B) = 540 Constraint 2: 540 = 22 *33 *5 The prime factors of 540 are 2, 3 and 5 So, the prime factors of LCM(A,B) are 2, 3 and 5 Checking the 3 pairs 108 and 5 The product of 108 and 5 is 540. So, the first Constraint is satisfied 108 = 2 *3 The prime factors of 108 are not 2, 3 and 5. So, Constraint 2 is not satisfied So, this pair is rejected 90 and 6 The product of 90 and 6 is 540. So, the first Constraint is satisfied 90 = 2*3 *5 The prime factors of 90 are 2, 3 and 5. So, Constraint 2 is also satisfied So, this pair is possible 27 and 20 The product of 27 and 20 is 540. So, the first Constraint is satisfied 27 = 33 The prime factors of 27 are not 2, 3 and 5. So, Constraint 2 is not satisfied So, this pair is rejectedLooking at the answer choices, we see that the correct answer is Option B