2008 iTest Problems/Problem 18

Problem

Find the number of lattice points that the line $19x+20y = 1909$ passes through in Quadrant I.

Solution

Solve for $y$ to get \[y = \frac{1909-19x}{20}\] In order for $y$ to be an positive integer, $1909-19x$ must be a multiple of 20 greater than $0$, so $x \le 100$. This means that the ones digit of $1909-19x$ is $0$ and the tens digit of $1909-19x$ is even.

The ones digit of $1909-19x$ is $0$ when the last digit of $x$ is $1$, so the available options are $1, 11, 21 \cdots 91$. However, since $1909-19x=1909-20x+x$, the tens digit must be odd. Thus, the only values that work are $11$, $31$, $51$, $71$, and $91$, so there are only $\boxed{5}$ lattice points in the first quadrant.

See Also

2008 iTest (Problems)
Preceded by:
Problem 17
Followed by:
Problem 19
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