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M04-31

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Bunuel
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M04-31 [#permalink] New post   16 Sep 2014, 00:23
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A perfect number is defined as a number for which the sum of all its unique positive factors, excluding the number itself, is equal to the number. For example, 6 is a perfect number because its factors, apart from 6 itself, are 1, 2, and 3, and their sum is \(1 + 2 + 3 = 6\). Which of the following numbers is also a perfect number?

A. 12
B. 20
C. 28
D. 48
E. 60
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C
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Bunuel
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Re M04-31 [#permalink] New post   16 Sep 2014, 00:23
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Official Solution:

A perfect number is defined as a number for which the sum of all its unique positive factors, excluding the number itself, is equal to the number. For example, 6 is a perfect number because its factors, apart from 6 itself, are 1, 2, and 3, and their sum is \(1 + 2 + 3 = 6\). Which of the following numbers is also a perfect number?

A. 12
B. 20
C. 28
D. 48
E. 60


Let's analyze each option by listing the factors from highest to lowest and adding them up as we go:

A. 12. Factors apart from 12 are: 6, 4, 3, ... STOP! The sum is already more than 12. Discard.

B. 20. Factors apart from 20 are: 10, 5, 4, 2, ... STOP! The sum is already more than 20. Discard.

C. 28. Factors apart from 28 are: 14, 7, 4, 2, and the last factor is 1. Their sum is 14 + 7 + 4 + 2 + 1 = 28. BINGO!

D. 48. Factors apart from 48 are: 24, 16, 12, ... STOP! The sum is already more than 48. Discard.

E. 60. Factors apart from 60 are: 30, 20, 15, ... STOP! The sum is already more than 60. Discard.

Among the given options, only 28 (C) is a perfect number, as the sum of its unique factors (excluding itself) equals the number.


Answer: C
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Re: M04-31 [#permalink] New post   04 Oct 2014, 10:57
How we have calculated Factor of 28 ? 4 and 14 are not prime .. is it correct to show factor as non prime number ?
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Re: M04-31 [#permalink] New post   04 Oct 2014, 11:11
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rahulgmat2014 wrote:
How we have calculated Factor of 28 ? 4 and 14 are not prime .. is it correct to show factor as non prime number ?


Factor of an integer is not necessarily a prime number.

A divisor of an integer \(n\), also called a factor of \(n\), is an integer which evenly divides \(n\) without leaving a remainder. In general, it is said \(m\) is a factor of \(n\), for non-zero integers \(m\) and \(n\), if there exists an integer \(k\) such that \(n = km\).

So, the factors of 28 are 1, 2, 4, 7, 14, and 28.

Theory on Number Properties: math-number-theory-88376.html
Divisibility tips: divisibility-multiples-factors-tips-and-hints-174998.html?hilit=divisibility%20tips

DS Divisibility/Multiples/Factors questions to practice: search.php?search_id=tag&tag_id=354
PS Divisibility/Multiples/Factors questions to practice: search.php?search_id=tag&tag_id=185[/textarea]

Hope it helps.
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Re: M04-31 [#permalink] New post   16 Nov 2014, 16:55
So when it says "unique factors' - that does not equal prime?
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Re: M04-31 [#permalink] New post   17 Nov 2014, 02:01
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phferr1984 wrote:
So when it says "unique factors' - that does not equal prime?


No, Unique factors = different factors.
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Re: M04-31 [#permalink] New post   05 Aug 2018, 06:47
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Is there a faster way to find ALL the factors of a number?
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Re: M04-31 [#permalink] New post   05 Aug 2018, 07:14
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imran1994 wrote:
Is there a faster way to find ALL the factors of a number?


Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

2. Properties of Integers




5. Divisibility/Multiples/Factors



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Re: M04-31 [#permalink] New post   30 Nov 2018, 08:23
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Bunuel wrote:
A perfect number is defined as one for which the sum of all the unique factors less the number itself is equal to the number. For instance, 6 is a perfect number, because the factors of 6 (apart from 6 itself) are 1, 2 and 3, and \(1 + 2 + 3 = 6\). Which of the following is also a perfect number?

A. 12
B. 20
C. 28
D. 48
E. 60

This is question is simple to calculate for all , To calculate sum of factors except the number , we calculate sum of all factors - the number.

Sum of all factors of a number is given by if a number is P= a^n * b^m * c^k. Sum of all factors of P = (1 + a ..... a^n) (1+b+b^2 .... b^m )(1+c + c^2+....c^k)

so 28 will have (1+ 2 + 4) (1+ 7) = 56 sum of factors ,excluding number will be 28, so the perfect number.
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Re: M04-31 [#permalink] New post   21 Apr 2023, 10:32
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Re: M04-31 [#permalink]
21 Apr 2023, 10:32
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