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The product of the proper positive integer factors of \(n\) can be written as \(n^{(ax+b)/c}\), where \(x\) is the number of positive divisors has, \(c\) is a positive integer, and the greatest common factor of the three integers \(a\), \(b\), and \(c\) is 1. What is \(a+b+c\)?

Guest Feb 12, 2019

#1**+1 **

**Here is one scenario that seems to work: Divisors of 24 =(1, 2, 3, 4, 6, 8, 12, 24) =8 divisors. proper product =(1, 2, 3, 4, 6, 8, 12)=13,824. Notice that I did not include 24 itself. 24^3 =13,824=24^(8*1 + 10) / 6, where: x=8, a =1, b=10 and c=6. GCD{1, 6, 10} = 1 a + b + c =1 + 10 + 6 = 17 Note: Also it seems to work for 60:**

**Divisors of 60 =(1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) = 12**

**Proper product( (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30) =777,600,000. Notice again I did include 60 itself.**

**60^5 =777,600,000 =60^(12*1 + 13) / 5, where:**

**x =12, a = 1, b = 13, and c =5**

**GCD, or GCF[1, 5, 13] = 1**

**a + b + c =1 + 13 + 5 = 19.**

**P.S. I believe there are infinite solutions to your question.**

Guest Feb 12, 2019

edited by
Guest
Feb 12, 2019