# Equation with two variables

1. Jan 9, 2014

### 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 alot 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.

2. Jan 9, 2014

### Dick

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.

3. Jan 10, 2014

### 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.