What is the sum of ALL the factors of 49?

Question

What is the sum of ALL the positive factors of 49?

A. 7
B. 8
C. 50
D. 51
E. 57

Kinshook · Answer

Asked: What is the sum of ALL the positive factors of 49?

Sum of all positive factors of 49 = 1 + 7 + 49 = 57

IMO E

itsprateek95 · Answer

Ans = E,

Two methods can be used for this

1) As the factors of 49 are quite few - 1,7,49 --> the sum = 57
2) General formula for sum of factors (can be used when there are a lot of factors) -->
Let n =  x^a * y^b * z^c , then the sum of factors for n is = (x^(a+1)-1)/(x-1) + (y^(b+1)-1)/(y-1) +(z^(c+1)-1)/(z-1) 
Hence, in this case 49 = 7^2; Therefore, formula =  (7^3 - 1) /(7-1) ---> 57

Fdambro294 · Answer

It’s a small enough number that you can list out the factors and then add them up to get the answer.

However in cases of larger numbers with several prime bases, the approach is as follows:

Ex: for the number 36 = (2)^2 * (3)^2, if we were to add up all the factors, we need to first understand the following:

Each unique positive factor can be “made” by using a unique combination of the prime factors.

For 2 we have as available options:
(2)^0 ——— (2)^1 ———(2)^2

For 3 we have as available options:
(3)^0 ———- (3)^1 ——— (3)^2

We can put together each unique factor by setting up all the different arrangements in which we pick 1 of the 3 available options for each prime base. Then we can SUM these factors:

(2)^0 * (3)^0 + (2)^0 *(3)^1 + (2)^0 *(3)^2

+ (2)^1 * (3)^0 + (2)^1 * (3)^1 + (2)^1*(3)^2

+ (2)^2 * (3)^0 + (2)^2 * (3)^1 + (2)^2 * (3)^2

Then for each line, we can take out the common factor or (2) to the given power:

(2)^0 *  
+
(2)^1 *  
+
(2)^2 *

Then you can take as common the expression

*

*   =   *   = 91

And

1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36 = 91

So to find the sum of all the unique positive factors of a given number:

(1st)Prime Factorize the number

(2nd) Add up all the Prime Base terms for each Prime Factor starting with the 0th Power (i.e., 1)

(3rd) Find the product of each of those SUMS to get the Sum of all the unique positive factors of the given number (note: can use the Sum of a Finite Geometric Sequence formula for larger amounts of bases)

For 49

49 = (7)^2

= sum of all positive factors

= 57

(E) 57

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Archit3110 · Answer

A perfect square has three factors 
Factors of 49 :1,7,49
Sum 1+7+49;57
Option E

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Anisha1637 · Answer

Factors of 49= 1, 7, 49
therefore their sum is 1+7+49= 57

What is the sum of ALL the factors of 49?

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What is the sum of ALL the factors of 49?

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