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

lwfrgin 2021-02-02 Answered
Determine the number of solutions of the congruence x461(mod117).
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Expert Answer

Caren
Answered 2021-02-03 Author has 96 answers
Step 1
Given: x461(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)31(mod9)
we deduce the congruence
x461(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)31(mod13)
So, the congruence x461(mod13)has(4,ϕ(13))=(4,12)=4 solutions
hence, the number of solutions of the congruence x461(d117)is2×4=8.
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