## Modular equation problem

9

I have problem solving this modular equation $67^n \equiv 67 \pmod {317026939759222841944}$ with $n>1$. I have tried my Laptop and Wolfram Alpha engine, but I don't get any solution, I'm very confused about this, can anyone out there help me ?

I usually can solve this kind of equation easily(only using ordinary scientific calculator), but this one is quite confusing.. – Anton GT – 2017-04-01T13:08:23.277

Can anyone give the value of n ?. – Anton GT – 2017-04-01T13:17:47.357

10

You are looking for the multiplicative order, MultiplicativeOrder[k, m]. The multiplicative order is the smallest exponent $k$ such that $x^k \equiv 1 \pmod m$.

Note that the modulus $m=317026939759222841944$ is divisible by prime $67$. Your equation then becomes $67^{n-1}\equiv 1 \pmod {4731745369540639432}$.

MultiplicativeOrder[67, 4731745369540639432]


11673080393745762

Check this solution with

PowerMod[67, 11673080393745762, 4731745369540639432]


1

and

PowerMod[67, 11673080393745762 + 1, 317026939759222841944]


67

Hence, the final answer is $n=11673080393745763$.