Say there are points on an elliptic curve $y^2=x^3+ax+b$ over $\mathbb{Z}_p$ and we want to express these points as $P=kQ$ for an integer $k$ and $P,Q$ are points on the curve.
I notice after $k=10$ that $P$ repeats itself. For example, when k=1 and k=11, we get the same P. This means that when $k=10$ then $Q$ is the point at infinity.
Each new point $P$ for $k>10$ can be represented as the point associated with $k \pmod{10}$.
How can I use modular arithmetic to prove this repeating pattern? Should it be a proof by induction?