To show that $((a,b), b) =(a,b)$

Question

I have to To show that $((a,b), b) =(a,b)$

Let $d=(a,b)$. So consider$(d,b)$. Now this means i have to find a greatest common divisor of (greatest common divisor of $a$ and $b$, and $b$. This is equivalent to saying to find gcd of a and b which is d. So i have $(d,b)=d$. Putting value of $d$ to get result back. is this correct ?

Answer 1

Lemma: if $k|d $ then $\gcd(k,d)=k $

Pf: $k $ is a common divisor of $k$ and $d$. If $m>k $ the $m\not \mid k $. So $k $ is greatest common divisor of $k $ and $d $.

So $\gcd (a,b) $ divides $b $ by definition. So $\gcd (\gcd (a,b),b)=\gcd (a,b)$.

Or simply note $\gcd((a,b),b) $ that $\gcd (a,b)$ is a common divisor of $\gcd (a,b)$ and $b $. So $\gcd( (a,b),b)\ge \gcd (a,b) $. But $\gcd( (a,b),b)$ divides $\gcd (a,b) $ so $\gcd (\gcd (a,b),b)\le \gcd (a,b) $ so $\gcd (\gcd (a,b),b)=\gcd (a,b) $.

Furthermore $\gcd (\gcd(a,b),b) $ divides $\gcd (a,b) $ which divides $a $ so $\gcd (\gcd (a,b),b) $ is a common divisor of $a $ and $b $ so...

Answer 2

Let $a=d\, a_1, b=d\, b_1$, $(a_1, b_1)=1$ so $(a,b)=d.$ Then $$ ((a,b),b)=(d, b)=(d, d b_1)=d (1,b_1)=d \cdot 1 =d =(a,b). $$