When 2 256 is divided by 17, the remainder would be : (CAT 2002) Options: A . &nbsp 1 B . &nbsp 16 C . &nbsp 14 D . &nbsp None of these E . &nbsp 81

When 2256 is divided by 17, the remainder would be : (CAT 2002)

Question

When $2^{256}$ is divided by 17, the remainder would be : (CAT 2002)

Answer: Option A
:
A

Ans: (a)

$2^{256}$ can be written as $(2^{4})^{64} = (17 - 1)^{64} .$

In the expansion of $(17 - 1)^{64}$ every term is divisible by $17$ except $(- 1)^{64}$ . Hence remainder is 1.

Or directly:

Euler's number of 17 is 16 and 256 is a multiple of 16, hence the remainder is 1.

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