Question
Find the remainder when $2^{100}$ is divided by 11 .
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Answer
We know that if $\mathrm{a} \equiv \mathrm{b}(\bmod \mathrm{m})$ and $0 \leq \mathrm{b} \leq \mathrm{m}$, then b is the remainder when a is divided by m . Therefore, to find the remainder when $2^{100}$ is divided by 11 , its is sufficient to find an integer $b$ such that $2^{100} \equiv b(\bmod 11)$, where $0 \leq b \leq 11$
Now,
$2^{1} \equiv 2(\bmod 11)$
$\Rightarrow 2^{2} \equiv 2 \times 2=4(\bmod 11)$
$\Rightarrow 2^{3} \equiv 2 \times 4=8(\bmod 11)$
$\Rightarrow 2^{4} \equiv 2 \times 8 \equiv 5(\bmod 11)\left[\because 2^{4} \equiv 16(\bmod 11)\right.$ and $\left.16 \equiv 5(\bmod 11) \therefore 2^{4} \equiv 5(\bmod 11)\right]$
$\Rightarrow 2^{5} \equiv 2 \times 5 \equiv 10(\bmod 11)$
$\Rightarrow 2^{5} \equiv-1(\bmod 11)[\because 10 \equiv-1(\bmod 11)]$
$\Rightarrow\left(2^{5}\right)^{20} \equiv(-1)^{20}(\bmod 11)$
$\Rightarrow 2^{100} \equiv 1(\bmod 11)$
Hence, 1 is the remainder when $2^{100}$ is divided by 11 .
Now,
$2^{1} \equiv 2(\bmod 11)$
$\Rightarrow 2^{2} \equiv 2 \times 2=4(\bmod 11)$
$\Rightarrow 2^{3} \equiv 2 \times 4=8(\bmod 11)$
$\Rightarrow 2^{4} \equiv 2 \times 8 \equiv 5(\bmod 11)\left[\because 2^{4} \equiv 16(\bmod 11)\right.$ and $\left.16 \equiv 5(\bmod 11) \therefore 2^{4} \equiv 5(\bmod 11)\right]$
$\Rightarrow 2^{5} \equiv 2 \times 5 \equiv 10(\bmod 11)$
$\Rightarrow 2^{5} \equiv-1(\bmod 11)[\because 10 \equiv-1(\bmod 11)]$
$\Rightarrow\left(2^{5}\right)^{20} \equiv(-1)^{20}(\bmod 11)$
$\Rightarrow 2^{100} \equiv 1(\bmod 11)$
Hence, 1 is the remainder when $2^{100}$ is divided by 11 .
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