Determine the number of solutions of the congruence x4≡61(mod117).

Step 1 Given: x4≡61(mod117) 117=32×13 As ϕ(9)4,ϕ(9)=64,6=62=3 (here(4,6) denotes the g.c.d of(4,6)) and (61)3≡(−2)3≡1(mod9) we deduce the congruence x4≡61(mod9)has(4,ϕ(9))=(4,6)=2 solutions Step 2 Similarity ϕ(13)4,ϕ(13)=124,12=124=3 and (61)3≡(−4)3≡1(mod13) So, the congruence x4≡61(mod13)has(4,ϕ(13))=(4,12)=4 solutions hence, the number of solutions of the congruence x4≡61(∓d117)is2×4=8.

"Determine the number of solutions of the congruence..." was answered by our community

lwfrgin

Answered question

2021-02-02

Determine the number of solutions of the congruence

x 4 \equiv 61 (mod 117)

Answer & Explanation

Caren

Skilled2021-02-03Added 96 answers

Step 1
Given:

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

117 = 3^{2} \times 13

\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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