$n$ is just a natural number. I've been scratching my head over this one for a while, so, I thought I'd seek a little help. My line of thinking is $8n + 3$ is always an odd number, which means, only an odd number can divide it. Thus the only overlapping divisors of $8n + 3$ and $5n + 2$ are odd numbers. But, I can't seem to make the next logical jump as to what this implies. Any help would be great. Thanks!
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1You can use the Euclidean algorithm on polynomials. I would start there. – ml0105 Mar 23 '14 at 22:43
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Hint $\ 8(5n+2)-5(8n+3) = 1 \ $ is divisible by any common divisor of $\ 5n+2,\ 8n+3.\, $ This Bezout Identity can be found by eliminating $\,n,\,$ or by the Extended Euclidean Algorithm.
For the latter, using the verson of the Extended Euclidean Algorithm described here yields
$$\begin{array}{rrr} 8n+3 & 1 & 0\\ 5n+2 & 0 & 1\\ 3n+1 & 1 & -1\\ 2n+1 &-1 & 2\\ n & 2 & -3\\ 1 &\color{#c00}{-5} & \color{#0a0}8\\ \end{array}\qquad\quad$$
where above lines $\,\ a\ \ b\ \ c\ \,$ mean $\ a = b(8n+3) + c(5n+2).\ $ So the Bezout identity is
$$ 1 \,=\, \color{#c00}{-5}(8n+3)+ \color{#0a0}{8}(5n+2)\quad $$
The linked post describes the algorithm in great detail, in a way that is easy to remember.
Bill Dubuque
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The bezout identity is gcd = ax + by. Now, if that's the case and the gcd is 1, then that's the only common divisor they share, correct? – zachary Mar 23 '14 at 23:00
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Okay, but, I'm not sure how to use the extended euclidean algorithm with those two equations. I googled it, but not luck. Do I just divide the two equations? – zachary Mar 23 '14 at 23:14
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