If s is the sum of all integers from 1 to 30, inclusive, what is the

Question

If s is the sum of all integers from 1 to 30, inclusive, what is the sum of all the factors of s?

(A) 303
(B) 613
(C) 675
(D) 737
(E) 768

amanvermagmat · Accepted Answer

Sum of integers from 1 to 30 = 30*31/2 = 15*31 = 3*5*31

Now there is a formula for calculating sum of all factors of a number. First we write a number in its 'prime factorised' form as:

N = a^p * b^q * c^r.. where a,b,c.. are distinct prime numbers in the base and p,q,r.. are their powers.. Then the formula for calculating sum of factors of N is:

Sum of factors of N = (1 + a + a^2 + ...+ a^p)(1 + b + b^2 +...+ b^q)(1 + c + c^2 +...+ c^r)..

Applying the same here, our number = 3*5*31
Required sum = (1+3)(1+5)(1+31) = 4*6*32 = 768. Hence  E answer

JaibeerSingh · Answer

Sum of all the integers 1 through 30 inclusive = number of terms multiplied by (first term plus last term) divided by 2.
So, s = 30x31/2 = 465.
Factors of 465 = 1,3,5,15,31,93,155,465.
Sum = 768.

Abhishek009 · Answer

s = \frac{(30 + 1)30}{2}

Or,  s = \frac{(30 + 1)15}{2}

Or,  s = 465

Factors of  465 = 3 * 5 * 31

So, sum of all the factors will be  (3+1)(5+1)(31+1) = 4*6*32 = 768

Thus, answer must be (E)

louisd8 · Answer

Hi Abhishek,

I don't understand why do you add one to each of the prime factors of 465 and then multiply them.

Besides that I find your methodology elegant and efficient!

Thanks

amanvermagmat · Answer

Hi Thats actually a formula to calculate the sum of factors of a number. I have explained that formula in this solution: https://gmatclub.com/forum/if-s-is-the- ... l#p1878412

ScottTargetTestPrep · Answer

The sum of all integers from 1 to 30, inclusive, is 30(30 + 1)/2 = 15(31) = 3 x 5 x 31 = 465. So s = 3^1 x 5^1 x 31^1 has (1 + 1) x (1 + 1) x (1 + 1) = 8 factors. These 8 factors are:

1, 465
3, 155
5, 93
15, 31

Therefore, the sum of all the factors of s is 1 + 3 + 5 + 15 + 31 + 93 + 155 + 465 = 768.

Answer: E

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If s is the sum of all integers from 1 to 30, inclusive, what is the

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