 # Determine the number of solutions of the congruence x4 -= 61 (mod 117). lwfrgin 2021-02-02 Answered
Determine the number of solutions of the congruence $x4\equiv 61\left(\text{mod}117\right)$.
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Step 1
Given: ${x}^{4}\equiv 61\left(\text{mod}117\right)$
$117={3}^{2}×13$
As $\frac{\varphi \left(9\right)}{4,\varphi \left(9\right)}=\frac{6}{4,6}=\frac{6}{2}=3$ (here(4,6) denotes the g.c.d of(4,6))
and ${\left(61\right)}^{3}\equiv {\left(-2\right)}^{3}\equiv 1\left(\text{mod}9\right)$
we deduce the congruence
${x}^{4}\equiv 61\left(\text{mod}9\right)has\left(4,\varphi \left(9\right)\right)=\left(4,6\right)=2$ solutions
Step 2
Similarity $\frac{\varphi \left(13\right)}{4,\varphi \left(13\right)}=\frac{12}{4,12}=\frac{12}{4}=3$
and ${\left(61\right)}^{3}\equiv {\left(-4\right)}^{3}\equiv 1\left(\text{mod}13\right)$
So, the congruence ${x}^{4}\equiv 61\left(\text{mod}13\right)has\left(4,\varphi \left(13\right)\right)=\left(4,12\right)=4$ solutions
hence, the number of solutions of the congruence ${x}^{4}\equiv 61\left(\mp d117\right)is2×4=8$.