27^400 mod 619
Solution:
`27^400" mod "619`
Here `27^400=(27^2)^200`
`=(27^2" mod "619)^200" mod "619`
`=(729" mod "619)^200" mod "619`
`=110^200" mod "619`
Here `110^200=(110^2)^100`
`=(110^2" mod "619)^100" mod "619`
`=(12100" mod "619)^100" mod "619`
`=339^100" mod "619`
Here `339^100=(339^2)^50`
`=(339^2" mod "619)^50" mod "619`
`=(114921" mod "619)^50" mod "619`
`=406^50" mod "619`
Here `406^50=(406^2)^25`
`=(406^2" mod "619)^25" mod "619`
`=(164836" mod "619)^25" mod "619`
`=182^25" mod "619`
Here `182^25=(182^2)^12*182`
`=(((182^2" mod "619)^12" mod "619)*(182" mod "619))" mod "619`
`=(((33124" mod "619)^12" mod "619)*182)" mod "619`
`=((317^12" mod "619)*182)" mod "619`
Here `317^12=(317^2)^6`
`=(((317^2" mod "619)^6" mod "619)*182)" mod "619`
`=(((100489" mod "619)^6" mod "619)*182)" mod "619`
`=((211^6" mod "619)*182)" mod "619`
Here `211^6=(211^2)^3`
`=(((211^2" mod "619)^3" mod "619)*182)" mod "619`
`=(((44521" mod "619)^3" mod "619)*182)" mod "619`
`=((572^3" mod "619)*182)" mod "619`
Here `572^3=(572^2)^1*572`
`=((327184" mod "619)*112)" mod "619`
`=(352*112)" mod "619`
`=39424" mod "619`
`=427`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
