Math Revolution and GMAT Club Contest! A perfect number n is defined

Question

Math Revolution and GMAT Club Contest Starts!

QUESTION #7:

A perfect number n is defined as the sum of all positive factors of integer n is 2n and also  f(m)=2^{m-1}(2^m-1)  is a perfect number, where m is an integer greater than 1. What is the difference of the second perfect number and the third perfect number?
 
A. 120 
B. 92 
C. 84 
D. 74 
E. 28

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mango aadmi · Answer

The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128

arthearoth · Answer

We're looking for the difference between the second and third perfect numbers, and we have two bits of information to tell us which numbers are perfect. Firstly the definition of a perfect number and then a formula to calculate perfect numbers.

I approached it cautiously as I wasn't sure whether they might redefine a perfect number for the purposes of this question or whether the two bits of information yielded the same list of numbers. It became clear as I worked through the question and using prior knowledge of perfect numbers that I was being over cautious.

Plugging in the second and third numbers allowed into the formula (1 is not allowed so 3 and 4 are the 2nd and 3rd value of m), gives us the second and third perfect numbers as 28 and 120. The difference between these two numbers is 92, so the answer is B.

NagrajB · Answer

The first perfect number is 6 when m = 2

The 2nd perfect number is 28 when m = 3

The 3rd perfect number is 120 when m = 4

120 - 28 = 92

kingjamesrules · Answer

The answer is B - 92.

Find the 2nd and 3rd perfect number that fits the following formula - f(m)=2^(m−1)(2^m−1). Here when m=2, f(m) turns out to be 6 which is the first perfect number. So now we have to find the 2nd and 3rd perfect numbers. Substitute m=3, f(m) = 28. Then go for m=4, f(m) = 120. So, 120-28=92. Hence answer is 92.

pluto82 · Answer

f(m) = 2^{m−1}(2^m−1)  is perfect number for m > 1
 f(2) = 2^{2−1}(2^2−1) = 2*3 = 6 
 f(3) = 2^{3−1}(2^3−1) = 4*7= 28 
 f(4) = 2^{4−1}(2^4−1) = 8*15= 120 
 f(5) = 2^{5−1}(2^5−1) = 16*31= 496

The difference of the second perfect number and the third perfect number
 f(4) - f(3) = 120-28 = 92

Answer (B)

AmoyV · Answer

m>1
The 1st perfect number is when m=2

2nd perfect number is when m=3
f(3)= 2^2(2^3-1) 
=4*7=28

3rd perfect number is when m=4
f(4)= 2^3(2^4-1) 
=8*15=120

120-28=92

Answer: B

donkadsw · Answer

OA is none of the options given.

first perfect number f(2) = 2^1*(2^2-1) = 6
second perfect number f(3) = 2^2*(2^3-1) = 28
third perfect number f(5) = 2^4*(2^5-1) = 496
 
So the difference between second and third perfect numbers is 496 - 28 = 458.

Kurtosis · Answer

f(m) = (2^(m - 1)) * ((2^m) - 1)

1st perfect number according to the formula: f(2) = 2 * 3 = 6
2nd perfect number according to the formula: f(3) = 4 * 7 = 28
3rd perfect number according to the formula: f(4) = 8 * 15 = 120

Difference = 120 - 28 = 92

Answer: B

shapla · Answer

QUESTION #7:
A perfect number n is defined as the sum of all positive factors of integer n is 2n and also f(m)=2m−1(2m−1) is a perfect number, where m is an integer greater than 1. What is the difference of the second perfect number and the third perfect number?

Solution:

f(2)=2X3=6
f(3)=2^2X7=4X7=28
f(4)=2^3X15=120

Differences between second and third perfect number =120-28=92

Answer:  B

HieuNguyenVN · Answer

With m=3, the second perfect number is:  2^2(2^3-1) = 4*7 = 28 
With m=4, the third perfect number is:  2^3(2^4-1) = 8*15 = 120 
(There is no other perfect number in between 28 & 120 because the the integer with total positive factor equals 2n is 496)
Then the difference between the third & the second perfect number is: 120 - 28 = 92.
Answer B.

DJ1986 · Answer

B) 92
I decided to pick B because all the f(m) numbers seem to alternate with either 8 or 6 so the difference between a number that is either an 8 or a 6 is going to end either with the digit 6 or 2. Only 92 ends with one of those.

Also, (2^(3-1))((2^3)-1)-(2^(4-1))(2^(4)-1) = (2^5-2^2)-(2^7-2^3) = 28-120 = -92

appsy01 · Answer

1st Perfect Number is 6 and is given by

f(2) = 2^1 (2^2 - 1) = 2 ( 4 - 1) = 2* 3 =6

Therefore, 2nd and 3rd Perfect Numbers are 
 f(3) = 2^2 (2^3 - 1) = 4 ( 8 - 1) = 4 *7 =28 
 f(4) = 2^3 (2^4 - 1) = 8 ( 16 - 1) = 8*15=120

120-28 = 92
Option B

rockthegmat007 · Answer

Using the formula starting m=2 (given m>1)

m=2 ; perfect number = 6 --> first perfect number 
m = 3 ; perfect number = 28 --> second perfect number 
m = 4 ; perfect number = 120 --> third perfect number

difference between third and second perfect number = 120 - 28 = 92 --> Choice B

HKD1710 · Answer

Placing the values of m in the formula would give us the perfect numbers in sequence:

1st perfect number (when m = 2) = 2*(4-1) = 6

2nd perfect number (when m = 3) = 4*(8-1) = 28

3rd perfect number (when m = 4) = 8*(16-1) = 120
 
difference of the 2nd and 3rd perfect number = 120-28 = 92 
  
Option B is the correct answer.

sri1sastra · Answer

2^m−1((2^m) −1), m>1
When m =2, it gives 6 as first perfect number.
When m =3, it gives 28 as second perfect number
When m=4, it gives 120 . This is not a perfect number.
When m =5, it gives 496 as the third perfect number.

Difference between third and second perfect number = 496-28 =468 . But this is not in answer choice.
Not sure if choices are correct.
Difference between third number(not perfect number, as m=4) and second perfect number = 120-28 =92. This is only in answer choice.So marking answer choice B to move on.

Icecream87 · Answer

A perfect number n is defined as the sum of all positive factors of integer n is 2n and also f(m)=2m−1(2m−1) is a perfect number, where m is an integer greater than 1. What is the difference of the second perfect number and the third perfect number?

A. 120 
B. 92 
C. 84 
D. 74 
E. 28

When m = 2, the first perfect number is 6 : test the perfect number  1+2+3+6 = 12  ok
m = 3, the second perfect number is 28 : test the perfect number  1^3 + 3^3 = 28  ok
m= 4 the answer is 120 is not a perfect number the sum of its factors = 360
m= 5 the third perfect number is 496 : 1^3 + 3^3 + 5^3 + 7^3  ok

Now I am stuck, i can't find my answer : 496 - 28 = 468
I will go with 120 - 28 = 92 answer B

Math Revolution and GMAT Club Contest! A perfect number n is defined

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