a perfect number?

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 amFollowed by: 80 members; Location: India; Thanked: 267 times; GMAT Score:760

by sanju09 » Sat Oct 08, 2011 4:20 am

A perfect number is defined as one that is equal to the sum of its distinct positive integer divisors that are less than the number itself. Which of the following is a perfect number?
(A) 24
(B) 25
(C) 28
(D) 30
(E) 35

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Source: Beat The GMAT — Problem Solving | Sat Oct 08, 2011 4:20 am

sl750: Master | Next Rank: 500 Posts; Posts: 496; Joined: Tue Jun 07, 2011 5:34 amFollowed by: 1 member; Thanked: 38 times

by sl750 » Sat Oct 08, 2011 4:25 am

Factors of 28

1,2,4,7,14 , Sum of the distinct factors = 28

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Sat Oct 08, 2011 4:28 am

sanju09 wrote:A perfect number is defined as one that is equal to the sum of its distinct positive integer divisors that are less than the number itself. Which of the following is a perfect number?
(A) 24
(B) 25
(C) 28
(D) 30
(E) 35

(A) 1 + 2 + 3 + 4 + 6 + 8 + 12 is not 24, FALSE
(B) 1 + 5 is not 25, FALSE
(C)1 + 2 + 4 + 7 + 14 = 28, TRUE

The correct answer is C.

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Abhishek009: Master | Next Rank: 500 Posts; Posts: 359; Joined: Wed Mar 11, 2009 4:37 amFollowed by: 2 members; Location: Kolkata, India; Thanked: 50 times

by Abhishek009 » Sat Oct 08, 2011 4:50 am

sanju09 wrote:A perfect number is defined as one that is equal to the sum of its distinct positive integer divisors that are less than the number itself. Which of the following is a perfect number?
(A) 24
(B) 25
(C) 28
(D) 30
(E) 35

All the known Perfect numbers can be found out using the simple formula -

All of the perfect numbers that have been found so far fit the formula

{2^(n-1) * ( 2^n ) - 1 }

Try plugging in the values of n , we can get the answer in seconds...

2^1 * ( 2^2 - 1 ) = 2 * 3 = 6

2^2 * ( 2^3 - 1 ) = 4 * 7 = 28

I didn't invent the formula here is the original formula -

https://mathforum.org/dr.math/faq/faq.pe ... l#allknown

Abhishek

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