Question

What is $2^{2^{2006}}(\bmod 3) ?$

Solution

VerifiedStep 1

1 of 2$\begin{align*} 2^{2^{2006}}\bmod 3=&2^{2\times2^{2005}}\bmod 3\\ =&\qty(4)^{2^{2005}}\bmod 3\\ =&\qty(4\bmod 3)^{2^{2005}}\tag{from (*)}\\ =&\qty(1)^{2^{2005}}=1\\ \end{align*}$

$(\text{We know that }(xy\bmod N)=(x\bmod N)\cdot( y\bmod N)\text{ (See problem 1.9)})\qquad(*)$

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