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## Homework Statement

Let [tex] a=p_{1}^{r_{1}}p_{2}^{r_{2}}...p_{k}^{r_{k}}, b=p_{1}^{s_{1}}p_{2}^{s_{2}}...p_{k}^{s_{k}} [/tex] where [tex] p_{1},p_{2},...,p_{k} [/tex] are distinct positive primes and each [tex] r_{i},s_{i} ≥ 0[/tex] Prove that [tex] (a,b)=p_{1}^{n_{1}}p_{2}^{n_{2}}...p_{k}^{n_{k}} \mbox{ where for each } i \mbox{, } n_{i}=\mbox{minimum of } r_{i},s_{i}. [/tex]

## Homework Equations

This is a question from a homework assignment from a first course in abstract algebra. The class has only been going for a few weeks. We've covered the long division algorithm for integers, the fundamental theorem of algebra, Euclid's algorithm and a few other menial theorems.

## The Attempt at a Solution

I've tried letting [tex] c=p_{1}^{h_{1}}p_{2}^{h_{2}}...p_{k}^{h_{k}}, h_{i}≥

0[/tex] be a divisor of a and b and then I just get lost. I'm not used to rigorous proof at all and this course has been a struggle for me while I get acquainted with the many styles of proof. Any sort of help or even guidance in the right direction would be much appreciated.