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

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Math Expert
Joined: 02 Sep 2009
Posts: 51229
If s is the sum of all integers from 1 to 30, inclusive, what is the  [#permalink]

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28 Jun 2017, 00:28
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5
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Difficulty:

55% (hard)

Question Stats:

64% (02:38) correct 36% (02:19) wrong based on 145 sessions

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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

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Posts: 46
GMAT 1: 700 Q49 V35
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Re: If s is the sum of all integers from 1 to 30, inclusive, what is the  [#permalink]

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28 Jun 2017, 07:35
1
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.
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Location: India
Re: If s is the sum of all integers from 1 to 30, inclusive, what is the  [#permalink]

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28 Jun 2017, 08:30
3
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
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Re: If s is the sum of all integers from 1 to 30, inclusive, what is the  [#permalink]

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28 Jun 2017, 10:29
2
Bunuel wrote:
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

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

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

Or, $$s = 465$$

Factors of $$465 = 3 * 5 * 31$$

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

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Joined: 18 Sep 2017
Posts: 4
Re: If s is the sum of all integers from 1 to 30, inclusive, what is the  [#permalink]

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07 Feb 2018, 04:36
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
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Joined: 21 Aug 2013
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Re: If s is the sum of all integers from 1 to 30, inclusive, what is the  [#permalink]

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07 Feb 2018, 04:42
louisd8 wrote:
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

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
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Re: If s is the sum of all integers from 1 to 30, inclusive, what is the  [#permalink]

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01 Aug 2018, 15:45
Bunuel wrote:
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

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.

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Re: If s is the sum of all integers from 1 to 30, inclusive, what is the &nbs [#permalink] 01 Aug 2018, 15:45
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