2024 AMC 10A Problems/Problem 18

Problem

There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base- $b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$ ?

$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

Solution

$2b^3+2b+4\equiv 0\pmod{16}\implies b^3+b+2\equiv 0\pmod 8$ , if $b$ even then $b+2\equiv 0\pmod 8\implies b\equiv 6\pmod 8$ . If $b$ odd then $b^2\equiv 1\pmod 8\implies b^3+b+2\equiv 2b+2\pmod 8$ so $2b+2\equiv 0\pmod 8\implies b+1\equiv 0\pmod 4\implies b\equiv 3,7\pmod 8$ . Now $8\mid 2024$ so $\tfrac38\cdot 2024=759$ but $3$ is too small so $758\implies\boxed{20}$ . ~OronSH ~mathkiddus

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2024 AMC 10A Problems/Problem 18

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