No Integral Solution for x & y when c Not Divisible by d

[kathrynag]

Homework Statement

For, a, b, c$$\in$$integers and D$$\in$$integers -{0}, if a and b are divisible by d, and c is not divisible by d then the equation ax+by=c has no integral solution for x and y.

Homework Equations

The Attempt at a Solution

ax+by=c
a/dx+b/dx=c
1/d[ax+by]=c
ax+by=cd

 

[HallsofIvy]

Where did the d in a/d come from? You can't just "stick" it into part of the equation.

Saying "d divides a" means a= dn for some integer n. Saying "d divides b" means b= dm for some integer m. Replace a and b in your equation by that and see what happens.

 

[kathrynag]

Where did the d in a/d come from? You can't just "stick" it into part of the equation.

Saying "d divides a" means a= dn for some integer n. Saying "d divides b" means b= dm for some integer m. Replace a and b in your equation by that and see what happens.

dnx+dmx=c
d(nx+mx)=c

 

[kathrynag]

I don't know if that was the correct way to do it and then maybe say since we don't have a dpc then there is no integral solution?