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

Question

Determine the number of solutions of the congruence

x^{4} \equiv 61 (m o d 117)

Answer 1

Step 1
Given: $x^{4} \equiv 61 (m o d 117)$
$117 = 3^{2} \times 13$
As $(ϕ (9)) / (4, ϕ (9)) = 6 / (4.6) = 6 / 2 = 3 (h e r e (4.6) d e n o t e s t h e g . c . d o f (4.6))$
and $(61)^{3} \equiv (- 2)^{3} \equiv 1 (m o d 9)$
we deduce the congruence
$x^{4} \equiv 61 (m o d 9) h a s (4, ϕ (9)) = (4.6) = 2 s o l u t i o n s$
Step 2
Similarity $y \frac{ϕ (13)}{4, ϕ (13)} = \frac{12}{4.12} = \frac{12}{4} = 3$
and $(61)^{3} \equiv (- 4)^{3} \equiv 1 (m o d 13)$
So, the congruence $x^{4} \equiv 61 (m o d 13) h a s (4, ϕ (13)) = (4.12) = 4 s o l u t i o n s$
hence, the number of solutions of the congruence $x^{4} \equiv 61 (m o d 117) i s 2 \times 4 = 8$ .

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

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