Number Theory. If d=gcd(a,b,c) then d is a linear combination of a,b, and c

[celtics777]

Homework Statement

Several of us claimed that if d=gcd(a,b,c) then d is a linear combination of a,b and c, i.e. that d=sa+tb+uc for some integers s,t, and u. That is true, but we only proved the analogous claim for the greatest common divisor of two numbers, i.e. when d=gcd(a,b). We need three.

Homework Equations

N/A?

The Attempt at a Solution

I know that gcd(a,b,c)=gcd(gcd(a,b)c). Can I use this to prove? If so, I'm not sure how.

 

[HallsofIvy]

Sure. If you know that gcd(x, y)= px+ qy, then gcd(gcd(a,b), c)= p(gcd(a,b))+ qc.
And gcd(a, b)= sa+ tb so gcd(gcd(a,b), c)= p(sa+ tb)+ qc= (ps)a+ (pt)b+ qc and ps, pt, and q are integers.