7^106 mod 143
Solution:
`7^106" mod "143`
Here `7^106=(7^3)^35*7`
`=(((7^3" mod "143)^35" mod "143)*(7" mod "143))" mod "143`
`=(((343" mod "143)^35" mod "143)*7)" mod "143`
`=((57^35" mod "143)*7)" mod "143`
Here `57^35=(57^2)^17*57`
`=(((57^2" mod "143)^17" mod "143)*(57" mod "143)*7)" mod "143`
`=(((3249" mod "143)^17" mod "143)*113)" mod "143` `("where "(57*7)" mod "143=399" mod "143=113)`
`=((103^17" mod "143)*113)" mod "143`
Here `103^17=(103^2)^8*103`
`=(((103^2" mod "143)^8" mod "143)*(103" mod "143)*113)" mod "143`
`=(((10609" mod "143)^8" mod "143)*56)" mod "143` `("where "(103*113)" mod "143=11639" mod "143=56)`
`=((27^8" mod "143)*56)" mod "143`
Here `27^8=(27^2)^4`
`=(((27^2" mod "143)^4" mod "143)*56)" mod "143`
`=(((729" mod "143)^4" mod "143)*56)" mod "143`
`=((14^4" mod "143)*56)" mod "143`
Here `14^4=(14^2)^2`
`=(((14^2" mod "143)^2" mod "143)*56)" mod "143`
`=(((196" mod "143)^2" mod "143)*56)" mod "143`
`=((53^2" mod "143)*56)" mod "143`
Here `53^2=(53^2)^1`
`=((2809" mod "143)*56)" mod "143`
`=(92*56)" mod "143`
`=5152" mod "143`
`=4`
This material is intended as a summary. Use your textbook for detail explanation.
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