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

Tyra 2020-12-25 Answered
Determine the number of solutions of the congruence x461(mod 117)
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Expert Answer

Pohanginah
Answered 2020-12-26 Author has 96 answers

Step 1
Given: x461(mod 117)
117=32×13
As (ϕ(9))/(4,ϕ(9))=6/(4.6)=6/2=3(here(4.6)denotes the g.c.d of(4.6))
and (61)3(2)31(mod 9)
we deduce the congruence
x461(mod 9) has (4,ϕ(9))=(4.6)=2 solutions
Step 2
Similarity yϕ(13)4,ϕ(13)=124.12=124=3
and (61)3(4)31(mod 13)
So, the congruence x461(mod 13) has (4,ϕ(13))=(4.12)=4 solutions
hence, the number of solutions of the congruence x461(mod 117) is 2×4=8.

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