For how many ordered pairs of positive integers $(x,y)$ is $x+2y = 100$?

Question

For how many ordered pairs of positive integers $(x,y)$ is $x+2y = 100$?

I'm thinking the answer is 16. Since it is x+2y=100 this implies that x is even. so x=2,4, 6, 8, 10, ..., 30, 32. There are 16ordered pairs.

Answer 1

We have $x=100-2y=2(50-y)$ and so $x>0$ iff $y<50$.

Therefore, every value of $y=1,\dots,49$ works.

Answer 2

Yes, $x$ would be even. And the value will range from $2, 4, 6,... 98$. We end at 98 because that would give the minimum possible value of $ y$ which is $1$. So in total, there are $49$ ordered pairs.