We're doing public key cryptography this week and I just can't seem to get a grasp on it.
I really don't know how to solve this problem. Can anyone point me in the right direction? I'd really appreciate it.
2026-03-30 00:19:13.1774829953
Determine a positive integer $e$ that satisfies $M^{17e}\equiv_{77}M$, when $(M,77)=1$.
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$\renewcommand{\phi}{\varphi}$You are done if you find $e$ such that $17 e \equiv 1 \pmod{\phi(77)}$. Here $\phi(77) = \phi(7) \phi(11) = 6 \cdot 10 = 60$ is Euler's $\phi$ function.
So run Euclid's algorithm to find $$ 17 \cdot (-7) + 60 \cdot 2 = 1, $$ so you can take $e = -7$, or if you prefer $e = -7 + \phi(77) = 53$.