Question

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

Answer 1

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 \(\displaystyle{x}^{{4}}\equiv{61}{\left(\mp{d}{117}\right)}{i}{s}{2}\times{4}={8}\).