# Determine the number of solutions of the congruence x^{4}equiv61(mod 117)

Question
Congruence
Determine the number of solutions of the congruence $$x^{4}\equiv61(mod\ 117)$$

2020-12-26
Step 1
Given: $$x^{4}\equiv61(mod\ 117)$$
$$117=3^{2}times 13$$
As $$(phi(9))/(4,phi(9))=6/(4.6)=6/2=3 (here (4.6)denotes the g.c.d of(4.6))$$
and $$(61)^{3}\equiv(-2)^{3}\equiv1(mod\ 9)$$
we deduce the congruence
$$x^{4}\equiv61(mod\ 9)\ has\ (4,\phi(9))=(4.6)=2\ solutions$$
Step 2
Similarity $$y\frac{\phi(13)}{4,\phi(13)}=\frac{12}{4.12}=\frac{12}{4}=3$$
and $$(61)^{3}\equiv(-4)^{3}\equiv1(mod\ 13)$$
So, the congruence $$x^{4}\equiv61(mod\ 13)\ has\ (4,\phi(13))=(4.12)=4\ solutions$$
hence, the number of solutions of the congruence $$x^{4}\equiv61(mod\ 117)\ is\ 2\times 4=8$$.

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