Can ax + by + cz = d have an integer solution?

[ACardAttack]

Prove that the Diophantine equation ax+by+cz+d has an integer solution if and only if the gcd(a,b,c) divides d.

Got this on my homework for my proofs class. Help would be greatly appreciated.

Thanks

 

[dodo]

Start with a simpler equation, ax+d, and then try to add the other terms one by one.

 

[ACardAttack]

Start with a simpler equation, ax+d, and then try to add the other terms one by one.

I goofed...it is actually ax+by+cz=d...does that make a difference?

 

[whodsow]

the Linear Equation Theorem says that the equation ax + by = gcd(a, b) always has a solution(s, u) in integers, and this solution can be found by the Euclidean algorithm, which we use to compute the gcd of a and b.

 

[ACardAttack]

I figured it out...thanks for the help

 

[HallsofIvy]

I goofed...it is actually ax+by+cz=d...does that make a difference?

Well, yes! It's an equation! ax+ by+ cz+ d isn't an equation.