Given that 13−1≡29(mod47), find 34−1(mod47), as a residue modulo 47. (Give a number between 0 and 46, inclusive.)
13−1≡29(mod47)(29)(13)≡1(mod47)(29)(47−34)≡1(mod47)(29)(47)+(−29)(34)≡1(mod47)−(29)(34)≡1(mod47)
34−1≡(−29)(mod47)(−29)≡(47−29)(mod47)(−29)≡18(mod47)34−1=18(mod47)
as a check18⋅34=612=13⋅47+118⋅34≡1(mod47)
