The remainder, when 2^100 is divided by 11, is :

The correct answer is Option (2) → 1

Since \( 2^{10}=1024 \) is one more than a multiple of 11:

    \( \displaystyle 2^{10}=1024=93\times 11+1 \)

    \( \displaystyle 2^{10}\equiv 1\ (\mathrm{mod}\ 11) \)

Raising both sides to the power 10:

    \( \displaystyle 2^{100}=(2^{10})^{10}\equiv 1^{10}=1\ (\mathrm{mod}\ 11) \)

So the remainder is 1.