Use definition!

For example by definition

1) if g is a factor of a and b and i

2) if d is another factor of a and b then g divides d.

since then divides both and . Hence divides both a and b.

Now supposed d divides and let q be any of prime factor of d. Since d divides a then q must be one of . Then the prime factors of d are also . So we can write where . By doing the same thing with respect to b, we also have . Hence . Then it follows that g divides d.

The same (similar) argument also works for lcm.