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A positive integer x is a perfect number if the sum of all the factors
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21 Sep 2021, 01:39
21 Sep 2021, 01:39
Expert Reply
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Question Stats:
58% (01:35) correct
41% (02:07) wrong based on 12 sessions
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A positive integer x is a perfect number if the sum of all the factors of x, including 1 and x, is equal to 2x.
A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
Source: manhattanreview
Quantity A |
Quantity B |
The sum of the reciprocals of all the factors of the perfect number 28 |
2 |
A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
Source: manhattanreview
Re: A positive integer x is a perfect number if the sum of all the factors
[#permalink]
21 Sep 2021, 06:55
21 Sep 2021, 06:55
1
Factors of 28 including 28 are 1, 2, 4,7,14,28 and taking inverse and adding them will give 2.
So Answer is C).
So Answer is C).
Re: A positive integer x is a perfect number if the sum of all the factors
[#permalink]
21 Sep 2021, 10:41
21 Sep 2021, 10:41
1
The factors of 28 are:
1,2,4,7,14,28
So the sum of the reciprocals are:
\(\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}\)
Let's put 2 on the right side of this, and figure out what sign should go between them (=,<,>, or undetermined). To denote the sign, I'll use \(?\).
\(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}\) \(?\) \(2\)
Let's multiply both sides by 28:
\(28(\)\(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28})\) \(?\) \(28(2)\)
The result is:
\(28 + 14 + 7 + 4 + 2 + 1\) \(?\) \(56\)
\(56\) \(?\) \(56\)
So we know that \(?\) is actually \(=\).
The answer therefore is C
1,2,4,7,14,28
So the sum of the reciprocals are:
\(\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}\)
Let's put 2 on the right side of this, and figure out what sign should go between them (=,<,>, or undetermined). To denote the sign, I'll use \(?\).
\(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}\) \(?\) \(2\)
Let's multiply both sides by 28:
\(28(\)\(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28})\) \(?\) \(28(2)\)
The result is:
\(28 + 14 + 7 + 4 + 2 + 1\) \(?\) \(56\)
\(56\) \(?\) \(56\)
So we know that \(?\) is actually \(=\).
The answer therefore is C
