Ball bouncing in a box, will it meet a vertex.

Question

I have no idea upon how to solve this:

A box 5cm by 3cm with a ball projected from a vertex at 45 degree angle, it reflexes at a 45 degree angle and keeps reflecting at a 45 degree angle. Will it meet a vertex and if so how many reflections before contact with vertex?

I have mental image the problem, but how to make that math?

Answer 1

We can unfold the trajectory of the ball and reflect the box instead. This gives us a $5\times 3$ grid in the plane and the question is whether the diagonal line through the origin will ever meet a lattice point again, and if so after crossing how many vertical or horizontal grid lines? Cast into a formula: Do there exist natural numbers $n,m$ with $(5n,3m)$ on the diagonal, i.e. with $5n=3m$? And what is the smallest such pair? An obvious candidate is $n=3$, $m=5$ because $5\cdot 3=3\cdot 5$, and in this case it is also the minimal solution because $\gcd(3,5)=1$. So the point itself is $(15,15)$ and the trajectory crosses $n-1$ vertical and $m-1$ horizontal lines until there (namely $x=5$, $x=10$, $y=3$, $y=6$, $y=9$, $y=12$). So the answer is: Yes, after six reflections.

Answer 2

To accompany Hagen's great response, here is a pictorial solution.