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

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3 00:00

Difficulty:   55% (hard)

Question Stats: 64% (02:03) correct 36% (01:51) wrong based on 114 sessions

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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

Source: www.GMATinsight.com

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

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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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Joined: 08 Jan 2019
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Re: If perfect numbers are defined as numbers whose all the factors sum up  [#permalink]

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In the case of 36: Is 6 considered as factor once or twice (as 6*6=36)?
CEO  D
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 3189
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
If perfect numbers are defined as numbers whose all the factors sum up  [#permalink]

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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.

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Joined: 04 Jan 2015
Posts: 3256
Re: If perfect numbers are defined as numbers whose all the factors sum up  [#permalink]

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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.

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