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 \(x^{4}\equiv61(mod\ 117)\ is\ 2\times 4=8\).

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 \(x^{4}\equiv61(mod\ 117)\ is\ 2\times 4=8\).