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Step 1 Given: x4≡61(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)3≡1(mod 9) we deduce the congruence x4≡61(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)3≡1(mod 13) So, the congruence x4≡61(mod 13) has (4,ϕ(13))=(4.12)=4 solutions hence, the number of solutions of the congruence x4≡61(mod 117) is 2×4=8.
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