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If perfect numbers are defined as numbers whose all the factors sum up

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New post 30 Jun 2017, 19:36
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Question Stats:

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If perfect numbers are defined as numbers whose all the factors sum up to twice the number then Which of the following is/are perfect number(s)?

I) 6
II) 28
III) 36

A) I only
B) II only
C) III only
D) I and II only
E) I, II and III all

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If perfect numbers are defined as numbers whose all the factors sum up  [#permalink]

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New post 30 Jun 2017, 22:00
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Definition : If sum of all factors is twice the number, then the number is a perfect number

Evaluating the options
I) 6 has 4 factors(1,2,3 and 6) - The sum of the factors is 12(twice the number 6)
Hence, this number is a perfect number,
II) 28 has 6 factors(1,2,4,7,14,28) - The sum of the factors is 56(twice the number 28)
Hence, this number is a perfect number
III) 36 has 9 factors(1,2,3,4,6,9,12,18,36) - The sum of the factors is 91(which is not twice the number 36)
Hence, this number is not a perfect number

I and II only are the perfect numbers, and Option D is the correct answer
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Re: If perfect numbers are defined as numbers whose all the factors sum up  [#permalink]

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New post 17 Jan 2019, 04:38
In the case of 36: Is 6 considered as factor once or twice (as 6*6=36)?
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If perfect numbers are defined as numbers whose all the factors sum up  [#permalink]

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New post 25 Jan 2020, 00:18
GMATinsight wrote:
If perfect numbers are defined as numbers whose all the factors sum up to twice the number then Which of the following is/are perfect number(s)?

I) 6
II) 28
III) 36

A) I only
B) II only
C) III only
D) I and II only
E) I, II and III all

Source: http://www.GMATinsight.com


Here is the video solution of this problem.



Answer: Option D
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Re: If perfect numbers are defined as numbers whose all the factors sum up  [#permalink]

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New post 25 Jan 2020, 12:21

Solution



Given
In this question, we are given that
    • Perfect numbers are defined as numbers whose all the factors sum up to twice the number

To find
We need to determine
    • Among 6, 28, and 36, which are the perfect number(s)

Approach and Working out
Factors of
    • 6 = 1, 2, 3, 6
    • 28 = 1, 2, 4, 7, 14, 28
    • 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36

Sum of factors of 6 = 1 + 2 + 3 + 6 = 12 = 2 x 6 (condition satisfied)
Sum of factors of 28 = 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 x 28 (condition satisfied)
Sum of factors of 36 = 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36 = 91 (condition not satisfied)

Thus, option D is the correct answer.

Correct Answer: Option D
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Re: If perfect numbers are defined as numbers whose all the factors sum up   [#permalink] 25 Jan 2020, 12:21
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