To calculate the sum of an integer's positive factors:
1. Prime-factorize the integer
2. Write the prime-factorization in the form (a^p)(b^q)(c^r)...
3. Sum of the factors = (1+a¹+ a²+a³+...+a^p)(1+b¹+ b²+b³+...+b^q)(1+c¹+ c²+c³+...+c^r)...
What is the sum of the positive factors of 600?
1. 600 = 2*2*2*3*5*5
2. 2*2*2*3*5*5 = 2³3¹5²
3. Sum of the factors = (1+2¹+2²+2³)(1+3¹)(1+5¹+5²) = 1860.
BTGmoderatorDC 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
For any set of consecutive integers:
Average = (biggest + smallest)/2
Sum = (count)(average)
For the integers 1 through 30, inclusive:
Average = (30+1)/2 = 31/2
s = Sum = (30)(31/2) = (15)(31)
To calculate the sum of the factors of s, apply the process discussed above:
1. (15)(31) = 3*5*31
2. 3*5*31 = 3¹5¹31¹
Sum of the factors = (1+3¹)(1+5¹)(1+31¹) = 768.
The correct answer is
C.
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