Find r and n in congruence $827. 3^{839} ≡ r \mod n$

Question

Edit: Find r and n in congruence $827. 3^{839} ≡ r \mod n$

I am new to this modulo arithmetic topic and was given a question to solve.

Find $n$ in the following equation: $$3^{839}\equiv827\bmod n$$

I have tried finding $3^{128}$ and thought of using the power laws to solve this equation but still unable to find $n$.

I am stuck and do not know how to proceed on any further.

Answer 1

839 is prime so using Fermat little theorem we can write:

$3^{839-1}≡1 \mod 839$

Multiplying both sides be 827 we get:

$827\times 3^{838}≡827 \mod 827\times 839≡827\mod 839$

So n can be equal to:

$n=839$ and $839\times 827=693853$