M04-31

Question

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

Show Answer

Official Answer

C

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

Bunuel · Answer

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

Show Answer

Official Answer

C

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

rahulgmat2014 · Answer

How we have calculated Factor of 28 ? 4 and 14 are not prime .. is it correct to show factor as non prime number ?

Bunuel · Answer

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 Hope it helps.

phferr1984 · Answer

So when it says "unique factors' - that does not equal prime?

Bunuel · Answer

No, Unique factors = different factors.

imran1994 · Answer

Is there a faster way to find ALL the factors of a number?

Bunuel · Answer

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  
     Theory   
 Math: Number Theory 
 Number Properties: Tips and hints 
 Advanced Number Properties on GMAT (All 5 parts) 
 Do you make these 3 mistakes in GMAT Even-Odd Questions?  
 Everything about Factorials on the GMAT 
 Solving Complex Problems Using Number Line 
     Questions   
 Tough New Set: Number Properties Set 
 Power of a Number in a Factorial Problems 
 Number Properties Ability Quiz 
 Special Numbers and Sequences Problems 
 The E-GMAT Number Properties Knockout: 10 Great 700+ Ques for you! 
 Trailing Zeros Problems 
 DS Questions 
 PS Questions

5. Divisibility/Multiples/Factors 
     Theory   
 Divisibility and Remainders on the GMAT 
 Tips and hints: Divisibility/Multiples/Factors 
 3 Deadly Mistakes you must avoid in LCM-GCD Questions 
 How Well Do You Know Your Factors? 
 THE MOST EFFICIENT WAY TO STUDY LEAST COMMON MULTIPLES ON THE GMAT 
     Questions   
 The E-GMAT Primes Trio - 3 Ques on No. of factors & Prime factors 
 DS Questions 
 PS Questions

For other subjects:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

gadde22 · Answer

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.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

bondi123 · Answer

I like the solution - it’s helpful.

kouassidrissa047 · Answer

I like the solution - it’s helpful.

Achillee · Answer

I like the solution - it’s helpful.

nobisaccusamus · Answer

I like the solution - it’s helpful.

crystxl · Answer

I like the solution - it’s helpful.

nissejosefsson · Answer

I like the solution - it’s helpful.

Condemned13 · Answer

I don’t quite agree with the solution. Poor example. I thought multiplying all factors results in the number 1*2*3=6. 
It should use a different example. Too easy to mistake with 1*2*3=6.

Bunuel · Answer

Please read carefully. The question explicitly says  sum , not product.

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

Show Answer

Official Answer

C

If you still read that as multiplication, that’s your problem, not a flaw in the example.

gmatshiva · Answer

I like the solution - it’s helpful.

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