Question

If dis the greatest common divisor of aand bin a Euclidean domain, prove that every associate of d is also a greatest common divisor of aand b.

Short answer

It is shown that every associate of d is also a greatest common divisor of a and b.

Step-by-step solution

Use Definition of Associate

By using definition of associate, we can say that let$\mathit{R}\mathbf{}$be a commutative ring with unit element. Two elements $\mathit{a}$and $\mathit{b}$in $\mathit{R}$is said to be associate if localid="1653283257133" $\mathit{b}\mathbf{=}\mathit{a}\mathit{u}$ for some localid="1653283445139" $\mathit{u}$ inR

Let $d$be the greatest common divisor of a and b

Also, suppose tis associate of d

$\Rightarrow t=du\Rightarrow d=t{u}^{-1}\Rightarrow t/d$

Prove t=gcd(a,b)

Now, d is greatest common divisor of a and b

Therefore, $d/a$and $d/b$

$\Rightarrow t/a$and$t/b[{·}_{·}·t/d]$

Now, if $x/a$and $x/b$for any$x\in R\Rightarrow x/d$

Now, $x/d$and$t=du\Rightarrow d/t$

Also, $x/d$and $d/t\Rightarrow x/t$for any$x\in R$

${·}^{·}·t/a$and$t/b$

For any $x\in R$such that$x/a$and$x/b$

$\Rightarrow x/t$

Therefore, $t=gcd(a,b)$.