A Passing Line

Assume the rectangular grid on the 2-dimensional Cartesian coordinate system, with the left lower corner at $(0,0)$ position. The line has the slope $\frac{13}{50}$ and the intersection of the line with each vertical line $x$ , happens at $\frac{13x}{50}$ . We know that at least one square from each column would be passed by the line (the answer would be at least $50$ ). The line would not pass more than $2$ squares from each column. If it does so, the slope of the line would be greater than $1$ (contradiction!). In order to find those column, in which two squares would contribute to the final answer, we need to find integer $0\leq x < 50$ , such that $\frac{13x}{50}-\lfloor \frac{13x}{50} \rfloor > \frac{37}{50}$ or, in other words, $\lceil\frac{13x}{50} \rceil - \frac{13x}{50} < \frac{13}{50}$ . To find such $x$ , we should see for which $x$

$13x \ (mod 50) > 37$

since $0 \leq x < 50$ and, $x\neq y \ , \ 0\leq x,y<50$ iff $13x \ (mod 50) \neq 13y \ (mod 50)$ , there are exactly $49-38+1=12$ such $x$ . so, the solution is $50+12=62$ .