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Find the remainder when 27^(40) is divid...

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

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`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.
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