Question

Find the number of solutions to the equation 2x + 5y = 2005, where x and y are positive integers

Answer 1

If y = 1, 2x = 2000 and x = 1000, so that works

if y = 2, 2x = 1995, so x is not an integer. Even integere values of y don't work.
If y = 399, 5y = 1995 2x = 10 and x = 5
If y = 400, 2x = 5 and x = 2.5
Higher values of y make x negative.
Only y values between 1 and 399 inclusive, that are odd integers, will satsify the requirement. How many is that?