Prove that given $a,b \in \mathbb{Z} : \gcd(a,b)=1 \Rightarrow \exists s,t : sa+tb=1$

Asked Dec 05 '16 at 20:19

Active Dec 05 '16 at 20:37

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(A) let $S=\{xa+yb: x, y \in \mathbb{Z}\}$. Explain why $S$ has a smallest positive element. Denote the smallest element by $u$

Attempted Solution: I tried plugging in values for a and b, since they can not be $0$ but the only common positive divisor is $1$. I assumed $1$ would be the only option for these values. I'm not sure if this is even what the question is asking.

edited Dec 05 '16 at 20:29

kingW3

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asked Dec 05 '16 at 20:19

Prisilla

$S\cap \Bbb N$ is a nonempty subset of $\Bbb N$, so it has a minimal element. – Dec 05 '16 at 20:30
Bezout's identity – kingW3 Dec 05 '16 at 20:34

Prove that given $a,b \in \mathbb{Z} : \gcd(a,b)=1 \Rightarrow \exists s,t : sa+tb=1$

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