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Question

What is 222006(mod3)?2^{2^{2006}}(\bmod 3) ?

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222006mod3=22×22005mod3=(4)22005mod3=(4mod3)22005=(1)22005=1\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*}

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

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