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# Ancient Number Theory Question

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Hey, so this is an old, like 2018 old, incorrectly and unanswered question. So I solved it myself. Here is the link to the old post: https://web2.0calc.com/questions/number-theory-help-divisor-arithmetic

Here is the question:

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

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?

Oct 11, 2020

Recall that by pairing up divisors of n, we can show that the product of the positive integer factors of n is $$n^{x/2}$$. We divide this formula by n to get the product of the proper positive integer factors of n, and we obtain $$\frac{n^\frac{x}{2}}{n} = n^{\frac{x}{2}-1} = n^\frac{x-2}{2}$$. Therefore, a=1, b=-2, and c=2, so a+b+c=1.