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

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

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?

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.

Shivam2612 · Answer

This is a great question that’s helpful for learning and I like the solution - it’s helpful.

M04-31

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