Inspired by Mohammad Rasel Parvej

Algebra Level 5

Let \(P(x) \in \mathbb{R}[x]\) be a Polynomial in x defined as follows: \(P(x) = x^{2016} + x^{2013} + x^{2010} + x^{2007} + ... + x^6 + x^3 +1\); \((P: \mathbb{R} \to \mathbb{R})\)

Let \(R(x)=C\) be the Polynomial that is obtained as remainder when \(P(x)\) is divided by \(x^3 -3\)

\(C\) is a Constant Natural Number. Find the number of \(1\)'s in Base-3(ternary base) in the representation of \(C\).


Bonus: How many 1's would have \(C\) if \(P : \mathbb{Z}_3 \rightarrow \mathbb{Z}_3\)?


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