Let and .
The idea is to show that and which implies .
It shouldn't be too troublesome to see why this is the case.
Question:
Assuming that the gcd(a,b) of any two integers a and b is defined, show that the gcd(a,b,c) of any three integers is also defined, and that (a,b,c)=((a,b),c).
My Answer so far: ( i think its totally wrong)
Let gcd(a,b)=d so d|b, d|a.
If gcd(a,b,c) = e exists then e|a , e|b and e|c ((1))
We want to prove that
gcd(a,b,c)=gcd(gcd(a,b),c)
=gcd(d,c)
=e
Since (a,b)=d and e|a and e|b => e|d
and from ((1)) e|c
As e|d and e|c we have gcd(d,c)=e
But i think i still need to show that (a,b,c) is definded and im not sure that i proved the statement correctly, i think i need to show that it is the greatest common divisor but i only showed that it was a divisor, im not sure. If anyone has a completly different solution that actually makes sense that would be super helpful!
Thanks