If n is a perfect square of a positive integer and is an integer, then a)divisible by 3b)divisible by 128c)divisible by 272 if n is a multiple of 3d)divisible by 107 if n is a multiple of 5e)None of the aboveCorrect answer is option 'C'. Can you explain this answer?

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If n is a perfect square of a positive integer and

is an integer, then

a)
divisible by 3
b)
divisible by 128
c)
divisible by 27² if n is a multiple of 3
d)
divisible by 10⁷ if n is a multiple of 5
e)
None of the above

Correct answer is option 'C'. Can you explain this answer?

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If n is a perfect square of a positive integer and is an integer, then...

Given

As is divisible by 2, that means n must be even. So, n must have 2 as one of its prime factors

To Find: To check the divisibility of

in the given options

Approach

Since n has 2 as one of its prime factors, we will find out the minimum possible power of 2 in the expression
Based on this deduction and the assumptions given in the options, we would check the options for their correctness.

Working Out

Since n is a perfect square, the power of every prime factor in n will be even. We’ve already established that 2 is a prime factor of n. So, n will at-least have 2²in it.
1. Therefore √n will have at-least 2¹in it and n² will at-least have 2⁴ in it.
2. Hence, the expression will definitely have 2⁶ = 64 in it. Let’s check the options now.

2. Evaluating the given options:

Divisible by 3 → We do not know if n has 3 as one of its prime factors. So, the expression may or may not be divisible by 3.
Divisible by 128 → The expression is definitely divisible by 64. It may or may not be divisible by 128.
Divisible by 27² if n is a multiple of 3 → If n has 3 as one of its prime factors, will definitely have 3⁷ in it. As

Now, we do not need to analyse further options as we have our answer in option C. However, we are doing it for the sake of completeness of the solution.
Divisible by 10⁷ if n is a multiple of 5→ If n is a multiple of 5, will definitely have 5⁷ in it. We also know that has 2⁶ in it. So, the expression must have 10⁶* 5 in it. So, it may or may not be divisible by 10⁷.

Answer: C

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If n is a perfect square of a positive integer and is an integer, then a)divisible by 3b)divisible by 128c)divisible by 272 if n is a multiple of 3d)divisible by 107 if n is a multiple of 5e)None of the aboveCorrect answer is option 'C'. Can you explain this answer?

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If n is a perfect square of a positive integer and is an integer, then a)divisible by 3b)divisible by 128c)divisible by 272 if n is a multiple of 3d)divisible by 107 if n is a multiple of 5e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If n is a perfect square of a positive integer and is an integer, then a)divisible by 3b)divisible by 128c)divisible by 272 if n is a multiple of 3d)divisible by 107 if n is a multiple of 5e)None of the aboveCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If n is a perfect square of a positive integer and is an integer, then a)divisible by 3b)divisible by 128c)divisible by 272 if n is a multiple of 3d)divisible by 107 if n is a multiple of 5e)None of the aboveCorrect answer is option 'C'. Can you explain this answer?.

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