Question

# Determine the number of solutions of the congruence x4 -= 61 (mod 117).

Similarity
Determine the number of solutions of the congruence $$\displaystyle{x}{4}\equiv{61}{\left(\text{mod}{117}\right)}$$.

2021-02-03
Step 1
Given: $$\displaystyle{x}^{{4}}\equiv{61}{\left(\text{mod}{117}\right)}$$
$$\displaystyle{117}={3}^{{2}}\times{13}$$
As $$\displaystyle\frac{{\phi{\left({9}\right)}}}{{{4},\phi{\left({9}\right)}}}=\frac{{6}}{{{4},{6}}}=\frac{{6}}{{2}}={3}$$ (here(4,6) denotes the g.c.d of(4,6))
and $$\displaystyle{\left({61}\right)}^{{3}}\equiv{\left(-{2}\right)}^{{3}}\equiv{1}{\left(\text{mod}{9}\right)}$$
we deduce the congruence
$$\displaystyle{x}^{{4}}\equiv{61}{\left(\text{mod}{9}\right)}{h}{a}{s}{\left({4},\phi{\left({9}\right)}\right)}={\left({4},{6}\right)}={2}$$ solutions
Step 2
Similarity $$\displaystyle\frac{{\phi{\left({13}\right)}}}{{{4},\phi{\left({13}\right)}}}=\frac{{12}}{{{4},{12}}}=\frac{{12}}{{4}}={3}$$
and $$\displaystyle{\left({61}\right)}^{{3}}\equiv{\left(-{4}\right)}^{{3}}\equiv{1}{\left(\text{mod}{13}\right)}$$
So, the congruence $$\displaystyle{x}^{{4}}\equiv{61}{\left(\text{mod}{13}\right)}{h}{a}{s}{\left({4},\phi{\left({13}\right)}\right)}={\left({4},{12}\right)}={4}$$ solutions
hence, the number of solutions of the congruence $$\displaystyle{x}^{{4}}\equiv{61}{\left(\mp{d}{117}\right)}{i}{s}{2}\times{4}={8}$$.