The number of solution pairs in positive integers of the equation 3x + 5y = 501 is:
- 33
- 34
- 35
- 100
- none of these
Here’s the graph. How many lattice points does it pass through?
Well, we could simply put y = (501 – 3x)/5 on a spreadsheet and count the number of integral values.
Or, if we begin with a calculator, we see that y is an integer when x = 2, 7, 12, 17, and so on.
This means that in every interval [0, 10], [11, 20], … , [160, 170], there are two values of x we can use, except in [160, 170] where we can’t use 167.
There are 16 such intervals, each with two good x‘s, plus 162.
That makes 2 x 16 + 1 = 33 good x‘s, thus 33 lattice points.