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
OA E
If s is the sum of all integers
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First let's find s. 1 to 30 inclusive is an evenly spaced set, so we know that the average = (high + low)/2 = (30 + 1)/2 = 15.5stevecultt 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
OA E
Sum = Average * Number of terms = 15.5 * 30 = 465
Prime factorization of 465 = 3 * 5 * 31
Now we can build all of 465's factors from those prime bases: 1, 3, 5, 3*5, 31, 3*31, 5*31, 465
Clean it up: 1, 3, 5, 15, 31, 93, 155, 465
Add 'em to get 768
(or simply see that the units' digit would be '8.'
The answer is E
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We have 30 consecutive integers starting from 1, thus the median of the numbers would be (1+30)/2 = 31/2. Since consecutive integers from 1 to 30 form an even spaced set, the median of the set = mean of the set.stevecultt 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
OA E
Thus sum s = Mean x number of integers = (31/2) x 30 = 15x31 = 3x5x31;
Thus, the prime factors of s are: 3, 5 and 31.
Thus, the factors of s are: 1; 3; 5; 31; 3 x 5 ƒ= 15; 3 x 31 ƒ= 93; 5 x 31 ƒ= 155; and 3 x 5 x 31 =ƒ 465.
Thus, the sum of all the factors of s ƒ 1 +‚ 3 +‚ 5 +‚ 31 +‚ 15 ‚+‚ 93 ‚+ 155 ‚+ 465 ƒ= 768.
The correct answer: E
Hope this helps!
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stevecultt wrote: ↑Mon Jun 12, 2017 4:20 amIf 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
OA E
The sum of all integers from 1 to 30, inclusive, is 30(30 + 1) / 2 = 15(31) = 3 x 5 x 31 = 465.
Recall that the total number of factors of a number is calculated by adding 1 to the exponent of each prime factor of the number and then multiplying those sums together. Since s = 3^1 x 5^1 x 31^1, then 465 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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