Discussion
What is the remainder when 2256 is divided by 17?
Answer: A
The remainder is 1.
Work with powers of 2 modulo 17. Since \(2^4 = 16 = 17 - 1\):
\[ \begin{aligned} 2^{4} &\equiv -1 \pmod{17} \\ 2^{8} &\equiv 1 \pmod{17} \\ 2^{256} &= (2^{8})^{32} \equiv 1 \pmod{17} \end{aligned} \]
16 would be the remainder for \(2^4\), or any power \(2^{4k}\) with \(k\) odd; 256 is a multiple of 8, so the result is 1. Fermat's little theorem gives the same, since \(2^{16} \equiv 1\) and 256 is a multiple of 16.
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