Question

Find the remainder when `27^(40)` is divided by 12.

Answer 1

`27^(40) = 3^(120)`
`3^(119) = (4-1)^(119)`
` = .^(119)C_(0)4^(119) - .^(119)C_(1)4^(118) + .^(119)C_(2)4^(117) - .^(119)C_(3)4^(116)+"….."+(-1)`
`= 4k - 1`
or `3^(120) = 12k - 3`
`= 12(k-1)+9`
Therefore, the required remainder is 9.