Greatest Common Devisor Proof

Question

I just need help structuring my proof correctly.

So the question states: Fix three natural numbers a,b,c and suppose that g= gcd (a,b) = gcd (a,c). Prove that gcd (b,c) $\geq$ g, and give an example where gcd (b,c) >g.

Here is what I have right now: a=2, b=4, c=8 because 2=gcd(2,4)=gcd(2,8) where g=2. Also, gcd(4,8)=4, and 4>2.

I also know that the definition of gcd is if you let a,b and c be integers, g|a, g|b, and g|c. Also, if c|a, and c|b, then |c| $\leq$ g.

Now I am just struggling to form a proper proof! Any help would be greatly appreciated. Thank you.

Answer 1

You have basically everything you need, so well done there. I will organize it below in a way that I find to be nice, but I must stress that your proof need not look like mine, and everyone develops their own style over time.

Let $a$, $b$, and $c$ be natural numbers, and suppose that $\gcd(a,b)=\gcd(a,c)=g$. Then by the definition of the greatest common divisor, we know that $g|b$ and $g|c$. Thus, $g$ is a common divisor of $b$ and $c$, and therefore $\gcd(b,c)\geq g$, again by definition.

Consider now the case $a=2$, $b=4$, and $c=8$. It is obvious that $\gcd(a,b)=\gcd(a,c)=2$. However, $\gcd(b,c)=4>2$, which shows that the inequality proven above need not be equality.

Answer 2

DEFINITION. $\gcd(x,y)=g$ if and only if

$(1.)\quad g \mid x \ \text{and} \ g \mid y$

$\text{$(2.) \quad$ If $z \mid x$ and $z \mid y$ then $z \mid g$}$

$ \ $

If $g \mid b$ and $g \mid c$ then ...

Try something simple for $a,b,$ and $c$. Like powers of $2$.