# Math Help - Divisibility (gcd) 4

1. ## Divisibility (gcd) 4

Show that: $(a,b)=(a,c)=1 \Rightarrow (a,b,c)=1$

2. Let $d = (a,b,c)$. This implies $d \mid a$, $d \mid b$, and $d \mid c$.

But this means $d$ is a common divisor of $a$ and $b$. So $d \mid (a,b)$.

Can you finish?

3. I think...

$(a,b)=1$ and $(a,c)=1$

$(a,b,c)=d \Rightarrow d|a ,d|b$ and $d|c \Rightarrow d|a, d|b$ and $d|a,d|c \Rightarrow d|(a,b)$and $d|(a,c)\Rightarrow d|1 \Rightarrow
d=1$

$\therefore$ $(a,b)=1$ and $(a,c)=1 \Rightarrow (a,b,c)=1$