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# Problem

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371
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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 n 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?

Jul 5, 2019

#1
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There maybe many solutions to this question:

Here is just one of them:

182 =2 x 7 x 13.   This can be written as : 182^[(2*3 + 7) /  13] = 182^1 =182

The GCD of  2, 7, 13 = 1

Sum = 2 + 7 + 13 = 22

Note: By "positive integer factors", I took it to mean "Prime integer factors".

Jul 5, 2019
edited by Guest  Jul 5, 2019
#2
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Here are many solutions between 1 and 100, for a, b, c:
a=1; b=1;c=(6 + a) / b; if(c==1 and isprime(b), goto4, goto5);print2," ", a," ", b," = ",2*a*b; a++;if(a<100, goto2, 0);a=1;b++;if(b<100, goto2, discard=0;

a   b    c          n
2   5   11     =  110
2   7   13    =  182
2   11   17   =  374
2   13   19  =  494
2   17   23  =  782
2   23   29  =  1334
2   25   31  =  1550
2   31   37  =  2294
2   35   41  =  2870
2   37   43  =  3182
2   41   47  =  3854
2   47   53  =  4982
2   53   59  =  6254
2   55   61  =  6710
2   61   67  =  8174
2   65   71  =  9230
2   67   73  =  9782
2   73   79  =  11534
2   77   83  =  12782
2   83   89  =  14774
2   91   97  =  17654

Jul 5, 2019