Integral Solution Set of Proofs: If int. soln's to ax+by+cz=e, then gcd(a,b,c) | e.

[goldenglove2000]

First, show that if there are integral solutions to the equation ax+by+cz=e, then gcd(a,b,c) divides e. Then, suppose that gcd(a,b,c) divides e. Show that there are integers w and z such that gcd(a,b)w +cz=e. Then, show that there are integers x and y such that ax+by=gcd(a,b)w

 

[goldenglove2000]

Integral Solution Proof

First, show that if there are integral solutions to the equation ax+by+cz=e, then gcd(a,b,c) divides e. Then, suppose that gcd(a,b,c) divides e. Show that there are integers w and z such that gcd(a,b)w + cz=e. Then, show that there are integers x and y such that ax+by=gcd(a,b)w.

My attempt:
Say that (a,b,c)=d. Since d|a, d|b, and d|c, then it follows that d|(ax+by+cz) for all integers x, y, and z. Thus, if ax+by+cz=e, then d|e, so (a,b,c)|e.

I did not know where to go from here and how to approach the other parts.

 

[Steven G]

First, show that if there are integral solutions to the equation ax+by+cz=e, then gcd(a,b,c) divides e. Then, suppose that gcd(a,b,c) divides e. Show that there are integers w and z such that gcd(a,b)w + cz=e. Then, show that there are integers x and y such that ax+by=gcd(a,b)w.

My attempt:
Say that (a,b,c)=d. Since d|a, d|b, and d|c, then it follows that d|(ax+by+cz) for all integers x, y, and z. Thus, if ax+by+cz=e, then d|e, so (a,b,c)|e.

I did not know where to go from here and how to approach the other parts.

What have you tried so far. Please show us that work. How are gcd(a,b) and gcd(a,b,c) related?