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Q. The remainder when $27^{40}$ is divided by $12$ is

Binomial Theorem

Solution:

$27^{40}= 3^{120}$
$3^{119} = (4-1)^{119} =\,{}^{119}C_04^{119} - \,{}^{119}C_1 4^{118}$
$+ \,{}^{119}C_24^{117} - \,{}^{119}C_34^{116}+ ...+ (- 1)$
$\therefore 3^{119} = 4k -1$
$\therefore 3^{120} = 12k - 3 = 12 (k- 1) + 9$
the required remainder is $9$.