Find the complete set of integer solutions in x and y to
821x + 1997y = 24047:
Determine all solutions with x > 0 and y > 0.
I started by finding the GCD(821, 1997)=1
Then I went backwards using the Euclidean algorithm and found 1=821(90)-1997(37).
So we have a particular solution xo=90, y0=-37.
Our particular solution is x0=24047*90=216630 and y0=24047*-37=-889,739.
then I have:
x=216,630+1997t and y=-889,739-821t.
Now it's the final part of determining all solutions with x>0 and y>0 that's getting me.
I'm a little confused on how you got those t values. I know it should be a pretty simple calculation.
Here's what I have done:
821x + 1997y = 24047
Use Division Algorithm to find GCD.
1997=821(2)+355
821=355(2)+111
355=111(3)+22
111=22(5)+1
22=1(22)+0
Then (821, 1997)=1 and 1|24047. So the equation is solvable.
1=111-22(5)
1=111-[355-111(3)]5
1=111(16)-355(5)
1=[821-355(2)]16-355(5)
1=821(16)-355(37)
1=821(16)-[1997-821(2)]37
1=821(90)-1997(37)
So 821x+1997y has a solution (90, -37) and thus 821x+1997y=24047 has a particular solution (2164230, -889739).
The general solution is given by:
x=2164230+1997t
y=-889739-821t
We have x>0.
Then 2164230+1997t>0
1997t>-2164230
t>-1083.74
We have y>0.
Then -889739-821t>0
-821t>889739
t<1083.72