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

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Bunuel
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If s is the sum of all integers from 1 to 30, inclusive, what is the [#permalink] New post   28 Jun 2017, 01:28
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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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E
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amanvermagmat
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Re: If s is the sum of all integers from 1 to 30, inclusive, what is the [#permalink] New post   28 Jun 2017, 09:30
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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] New post   28 Jun 2017, 11:29
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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 \((3+1)(5+1)(31+1) = 4*6*32 = 768\)

Thus, answer must be (E)
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Re: If s is the sum of all integers from 1 to 30, inclusive, what is the [#permalink] New post   28 Jun 2017, 08:35
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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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Re: If s is the sum of all integers from 1 to 30, inclusive, what is the [#permalink] New post   01 Aug 2018, 16:45
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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.

Answer: E
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Re: If s is the sum of all integers from 1 to 30, inclusive, what is the [#permalink] New post   07 Feb 2018, 05:36
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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amanvermagmat
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Re: If s is the sum of all integers from 1 to 30, inclusive, what is the [#permalink] New post   07 Feb 2018, 05:42
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] New post   05 Sep 2023, 13:37
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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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