How do I show this? I have an idea of what to do, but the problem overall is a little confusing to me. I can start the problem, but I just do not see how to get to the solution.
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1What do you know about numbers that are relatively prime? – paw88789 Oct 20 '14 at 01:58
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Their only common factor is 1. – JCMcRae Oct 20 '14 at 01:59
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1Do you know anything about linear combinations and gcd? – paw88789 Oct 20 '14 at 02:00
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Yes. So then, their GCD is 1, and by Euclidean Algorithm, there exists numbers s and t such that sa + bt = 1. – JCMcRae Oct 20 '14 at 02:01
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Hints:
From the given information, $c=aj$ and $c=bk$ for some $j,k\in\mathbb{Z}$. And $am+bn=1$ for some $m,n\in\mathbb{Z}$.
Therefore $c=amc+bnc$.
Can you take it from here? You need to get $c = ab\cdot(\text{integer})$
paw88789
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1I think so. So check my math here:
c = amc + bnc.
If we subsitute the values for c, we get c = am(bk) + bn(aj). Next, we subsitute out an ab, so c = ab(mk+nj)?
– JCMcRae Oct 20 '14 at 02:14 -