Determine the number of solutions of the congruence x4 -= 61 (mod 117).

Question

Determine the number of solutions of the congruence

x 4 \equiv 61 (mod 117)

.

Answer 1

Step 1
Given:

x^{4} \equiv 61 (mod 117)

117 = 3^{2} \times 13

As

\frac{ϕ (9)}{4, ϕ (9)} = \frac{6}{4, 6} = \frac{6}{2} = 3

(here(4,6) denotes the g.c.d of(4,6))
and

{(61)}^{3} \equiv {(- 2)}^{3} \equiv 1 (mod 9)

we deduce the congruence

x^{4} \equiv 61 (mod 9) h a s (4, ϕ (9)) = (4, 6) = 2

solutions
Step 2
Similarity

\frac{ϕ (13)}{4, ϕ (13)} = \frac{12}{4, 12} = \frac{12}{4} = 3

and

{(61)}^{3} \equiv {(- 4)}^{3} \equiv 1 (mod 13)

So, the congruence

x^{4} \equiv 61 (mod 13) h a s (4, ϕ (13)) = (4, 12) = 4

solutions
hence, the number of solutions of the congruence

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

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