Question

Let a, b, c be any three integers. Define gcd(a, b, c), the gcd od a, b, c as a positive integer d such that
(i) d|a, d|b, d|c and
(ii) if f|a, f|b, f|c then f|c
prove that gcd(a, b, c) =gcd(a, (b,c)) =gcd ((a,b),c) = gcd ((a,c),b)

Answer 1

Using the definition,

$$\operatorname{gcd}(b, c) = d$$

such that $d|a; d|b$ and if $f|a, f|b$ then $f|d$

So

$$\operatorname{gcd}(a, \operatorname{gcd}(b, c)) = \operatorname{gcd}(a, d) = e$$

Again, using the definition, $e|a$ and $e|d$. Since $d|a$ and $d|b$ we get $e|a$ and $e|b$. So $e|a; e|b; e|c$.

Next, if $f|a$ and $f|d$ then $f|e$. Using the fact $d|b$ and $d|c$ then we have implication

$$f|a, f|b, f|c \Rightarrow f|e$$

So

$$\operatorname{gcd}(a, \operatorname{gcd}(b, c)) = \operatorname{gcd}\big(a, (b, c)\big) = \operatorname{gcd}(a, b, c)$$

Using the symmetry

$$\operatorname{gcd}(a, b, c) = \operatorname{gcd}\big(a, (b, c)\big) = \operatorname{gcd}\big((a, b), c\big) = \operatorname{gcd}\big((a, c), b\big)$$