why $x^3=11^3$ mod $5083$ has only one solution?

Question

Why $x^3=11^3$ mod $5083$ has only one solution? (The only answer is $x=11$)

I know it has at most 3 roots but how to find that there isn't another answer?

Answer 1

As $5083=13\cdot 17\cdot 23$

We have $\displaystyle\left(\frac x{11}\right)^3\equiv1\pmod{13}\ \ \ \ (1)$

As all primes have primitive roots, taking discrete logarithm wrt some primitive root $g$ of $13,$(say $2$)

$3$ind$\displaystyle_g \left(\frac x{11}\right)\equiv0\pmod {12}\ \ \ \ (2)$

Using Linear Congruence Theorem, $(2)$ hence $(1)$ have $(3,12)=3$ solutions namely, $4k\pmod{13}$ where $k=0,1,2$ of $(2)$

$\implies 2^{4k}\pmod{13},k=0,1,2$ of $(1)$

Similarly, $\displaystyle\left(\frac x{11}\right)^3\equiv1\pmod{17}$ and $\displaystyle\left(\frac x{11}\right)^3\equiv1\pmod{23}$ should have unqiue solutions namely $1$

Using Chinese Remainder Theorem, we shall get three in-congruent solutions $\pmod{13\cdot 17\cdot 23}$