math
posted by Candice on .
Find the number of solutions to the equation 2x + 5y = 2005, where x and y are positive integers.
Thank u :)

this is an example of what is called a Diophantine equation.
There are several ways to solve these, I use a method involving continued fractions, but it is too messy to explain here.
Let's use a "common sense" method.
Consider the intercepts,
We have (0,401) and (1002.5,0) but neither of these would be allowed since you are looking for positive integers.
The closest point to the yintercept that would work is
(1000,1)
Now the slope of the line is 2/5, which means that for every increase of 2 for a y, we need a decrease of 5 for the x.
so possible points would be
(995,3), (990,5), (985,7) etc. until we get close to the xintercept
We could then have 1000/5 or 401 "jumps" to get to (5,399)
so the points from smallest x to largest x would be
(5,399), (10,397) ... (995,3), (1000,1)
so there are 401 points which are solutions, using only positive integers.