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In Pinter's book it states that the greatest common divisor of a and b can be written as the linear combination:

$t=la + kb $ for some integers $l$ and $b$

What is the proof of this?

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  • $\begingroup$ Look up the "Euclidean Algorithm" $\endgroup$ – Nick Mar 17 '16 at 20:08
  • $\begingroup$ Look up the "Extended Euclidean Algorithm" $\endgroup$ – mvw Mar 17 '16 at 20:09
  • $\begingroup$ I see how that works but how does it relate to this? $\endgroup$ – Augs Mar 17 '16 at 20:40
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This is standard bookwork. Take the smallest positive $la+kb$ for integers $l,k$. Call it $d$. Suppose it does not divide $a$. Then we have $a=qd+r$ with $0<r<d$, but $r$ is a linear combination of $a,b$. Contradiction. Similarly $d$ must divide $b$ and hence $t$. But $t$ obviously divides $d$, so they are equal.

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  • $\begingroup$ What do you mean smallest possible integers? 0? $\endgroup$ – Augs Mar 17 '16 at 20:27
  • $\begingroup$ Sorry I meant positive. Autocorrect $\endgroup$ – Augs Mar 17 '16 at 20:37
  • $\begingroup$ Also if l and k are positive won't t be bigger than a and b? How can the common divisor be bigger that either a or b? $\endgroup$ – Augs Mar 17 '16 at 20:42
  • $\begingroup$ Where did I say they were positive? Indeed you have just established that only one of $k,l$ will be positive! $\endgroup$ – almagest Mar 17 '16 at 21:30
  • $\begingroup$ Ah you were talking about the whole sum being positive $\endgroup$ – Augs Mar 17 '16 at 21:38

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