2023 AIME II Problems/Problem 15
Solution
Denote . Thus, for each , we need to find smallest positive integer , such that \[ 23 b_n = 2^n k_n + 1 . \]
Thus, we need to find smallest , such that
\[
2^n k_n \equiv - 1 \pmod{23} .
\]
Now, we find the smallest , such that . We must have . That is, . We find .
Therefore, for each , we need to find smallest , such that \[ 2^{{\rm Rem} \left( n , 11 \right)} k_n \equiv - 1 \pmod{23} . \]
We have the following results: \begin{enumerate} \item If , then and . \item If , then and . \item If , then and . \item If , then and . \item If , then and . \item If , then and . \item If , then and . \item If , then and . \item If , then and . \item If , then and . \item If , then and . \end{enumerate}
Therefore, in each cycle, , we have , , , , such that . That is, . At the boundary of two consecutive cycles, .
We have . Therefore, the number of feasible is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)