What is the smallest positive integer n such that 3n≡1356(mod22)?
What is the smallest positive integer n such that 3n≡1356(mod22)
3n≡1356(mod22)3n≡1356−61∗22(mod22)3n≡14(mod22)n≡143(mod22)n≡14∗3−1(mod22)3−1(mod22)=3ϕ(22)−1(mod22)|ϕ(22)=22∗(1−12)∗(1−111)=10=310−1(mod22)=39(mod22)=19683(mod22)=19683−894∗22(mod22)=15(mod22)3−1(mod22)=15(mod22)n≡14∗15(mod22)n≡210(mod22)n≡210−9∗22(mod22)n≡12(mod22)
The smallest positive integer n is 12