2025 AIME I 第 1 题

Write the two numbers in base $b:$ they are $b + 7$ and $9b + 7$

Since $9(b + 7) - (9b + 7) = 56,$ the condition is that $b + 7$ divides $56$

In base $b$ the two numbers are $17_b = b + 7$ and $97_b = 9b + 7.$ We need $9b + 7$ to be divisible by $b + 7,$ and since $b + 7$ certainly divides $9(b + 7) = 9b + 63,$ this is equivalent to $$\begin{gathered} (9b + 63) - (9b + 7) \\ = 56 \end{gathered}$$ being divisible by $b + 7.$

For $b \gt 9$ we have $b + 7 \gt 16,$ so $b + 7$ must be $28$ or $56,$ giving $b = 21$ or $b = 49.$ The sum is $21 + 49 = 70.$