If the product of two positive integers is 54...

### Related Test 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 PRASAD 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 View courses related to this question Explore GMAT courses
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