How can I simplify finding positive odd solutions to the equation 17x+11y=1000?

[MSG100]

Problem:
Find all the positive integer solutions where x and y are odd numbers, to the equation: 17x+11y=1000

Attempt of solution:

First attempt:
With Diophantine equation have gotten the answers:

x=2000
y=-3000

and the general solutions will be:

x=2000-11k
y=-3000+17k

Now I don't know what to do.

Second attempt:
If I skip the Diophantine solution and do it like this:

y=(1000-11x)/17

Now I see that x has to be in the interval 0≤x≤58 if y should be positive.

If I test all the odd numbers in the interval I'll get 3 solutions when both x and y are positive and odd numbers. The solutions are:

(x, y) = (9, 77), (31, 43) and (53, 9)


This solutions (which should be the right answer) takes a lot of time because you have to test all odd numbers between 0 to 58 (29 different numbers).

I need help to find an easier solution.

 

[Dick]

Problem:
Find all the positive integer solutions where x and y are odd numbers, to the equation: 17x+11y=1000

Attempt of solution:

First attempt:
With Diophantine equation have gotten the answers:

x=2000
y=-3000

and the general solutions will be:

x=2000-11k
y=-3000+17k

Now I don't know what to do.

Second attempt:
If I skip the Diophantine solution and do it like this:

y=(1000-11x)/17

Now I see that x has to be in the interval 0≤x≤58 if y should be positive.

If I test all the odd numbers in the interval I'll get 3 solutions when both x and y are positive and odd numbers. The solutions are:

(x, y) = (9, 77), (31, 43) and (53, 9)


This solutions (which should be the right answer) takes a lot of time because you have to test all odd numbers between 0 to 58 (29 different numbers).

I need help to find an easier solution.

If you don't want to check all those numbers you should use your Diophantine solution. You just have to figure out what values of k will make both x and y positive. There aren't that many.

 

[MSG100]

Thanks, that makes sense!

Then I just have following numbers k= 177, 178, 179, 180, 181 to make x and y positive and only k= 177, 179, 181 to make them positive AND odd.