Check out my blog post
http://gmatquant.blogspot.com/2010/11/integral-solutions-of-ax-by-c.html
in which I discussed in detail how find integral solutions of linear equations in 2 variables.
Once you know how to solve any such equation, you will know what I am talking about below:

4x -17y = 1
First solution: I check for y = 1. Do I have a multiple of 4 which is 1 greater than 17? Nope. I move on to y = 2. Do I have a multiple of 4 which is 1 greater than 34? Nope. What about if y = 3? 17y = 51 which is 1 less than 52 (4 x 13).

So first solution is x = 13 and y = 3
Next solution will be obtained by adding 17 to x and adding 4 to y.
Second solution is x = 30 and y = 7
Next solution will be obtained by adding 17 to x and adding 4 to y
Third solution is x = 47 and y = 11
and so on ...
There are infinite solutions but we have constraints.
x <= 1000 and x and y should be positive.

We have to find how many terms are there in the sequence 13, 30, 47, ... (terms less than 1000)
All numbers are of the form 13 + 17a.
You want to find the greatest such number less than 1000. Divide 1000 by 17. You get 14 remainder. So 986 (=17 x 58) is the greatest number less than 1000 divisible by 17. When you add 13 to it, you get 999. So 999 is the greatest number of this sequence.

In the first term in the sequence above, a = 0, in the last term, a = 58. Hence number of such solutions = 59.
_________________

Karishma
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Karishma, that was a very helpful blog post. Thank you!

We are asked to find integer solutions for x,y.

There is no need to apply Euclidian algorithm here, we can get particular solutions simply by plugging in some values.
As y reaches 3 we get:

y’ = 3 and x’ = 13. We need only to solve for x.

General solution for x will be:

x = x’ + bn = 13 + 17n

Hence lower limit for n is n ≥ 0.

Now we’ll find upper limit using given restriction x ≤ 1000. Plugging in general solution we’ll get:

13 + 17n ≤ 1000
17n ≤ 987
n ≤ 58

And our list of possible values of n has following representation:

0, 1,2, 3, ….. , 58

Total number of elements is 59.

KarishmaB can you plase explainhow did you figure out that next solution will be obtained by adding 17 to x and adding 4 to y.?
thanks