remainder

27⁴⁰
= (24 + 3)⁴⁰
= 12k₁ + 3⁴⁰
= 12k₁ + (3⁵)⁸
= 12k₁ + 243⁸
= 12k₁ + (240 + 3)⁸
= 12k₁ + 12k₂ + 3⁸
= 12k₃ + (3⁴)²
= 12k₃ + 81²
= 12k₃ + (72 + 9)²
= 12k₃ + 12k₄ + 9²
= 12k + 81

Hence,

remainder of 27⁴⁰ / 12 = remainder of (12k + 81) / 12 = 9

http://targetiit.com/iit_jee_articles/remainder_of_division_an_interaction_41.html

yeh dekh lo u'll get ur answer

hey how did you guess that?!! i mean how did you get that it was going to happen this way ?

like first 24+3 (that's fine i too did it)
but 243 and then 240+3 was something which would not have striken me if i were in the examination hall

You have to split it in such a way that you get 12p + something.Then continu the process until that something is less than 12.Then this is your ans.

Usually it's better you try to make that something=1 or 10ⁿ(if possible) so that your work is simplified.