Hey! the question is:

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).

This is what i have done:

let d1 = ((a,b),c)

where (a,b) = gcd(a,b) and is defined

hence let e = (a,b)

thus, d1 = (e,c)

since this is the gcd of two numbers, then the gcd(e,c)is also defined.

hence, d1 = ((a,b), c) is defined

now let d = (a,b,c)

d = ax0 +by0 +cz0

let a = d1f, b = d1g, c = d1h

then,

d = d1fx0 + d1gy0 + d1hz0

= d1(fx0 + gy0 + hz0)

therefore, d | d1

i know i need to show that d=(a,b,c) = ((a,b),c)=d1, and dat for this to occur d|d1 and d1|d, but im not sure if what i have done is right especially this part:

d = ax0 +by0 +cz0

let a = d1f, b = d1g, c = d1h

then,

d = d1fx0 + d1gy0 + d1hz0

= d1(fx0 + gy0 + hz0)

therefore, d | d1

Plus i also dont know how to continue -_-'.