My problem is the following:
$E$ is an elliptic curve $y^2 = x^3 + bx + c$ over integers modulo $221 = 13\cdot 17$.
There exist some points $P$ and $Q$ on $E$ such that $11P = \mathcal{O}$ and $7Q = \mathcal{O}$.
Can you determine $\sharp E$, the number of points on $E$?
What I've noted/tried:
The order of $P$ is $11$ and the order of $Q$ is $7$.
This looks an awful lot like Schoof's Algorithm. If I could use the Chinese Remainder Theorem to combine the results, how? What would I do with the result? I'm guessing something to do with $7\cdot 11 > 4\sqrt{221}$
I figured out the problem:
Recall Lagrange's Theorem, which states that the order of any element in a group divides the number of elements in the group. Thus, $7 \mid \sharp E$ and $11 \mid \sharp E$. Therefore, $7\cdot 11 = 77 \mid \sharp E$.
By Hasse's Theorem, $\sharp E$ lies in the range $(221+1-2\sqrt{221},221+1+2\sqrt{221}) \approxeq (192,252)$. Thus, $\sharp E = 3 \cdot 77 = 231$.