P16260 [Lanqiao Cup 2026 NOI Qualifier Python/Java Group B] Peaceful Mealtime Day

Cat owner Xiao Lan keeps $26$ huge Maine Coon cats at home. To make their lives more fun, Xiao Lan bought a “weird feeder” with artificial intelligence. This feeder has a very user-unfriendly rule for dispensing food: the total number of cat food pellets it spits out each day is an integer formed by concatenating the natural numbers starting from $1$ in order. Specifically: - On day $1$, the feeder spits out $1$ pellet; - On day $2$, the feeder spits out $12$ pellets (concatenating $1$ and $2$); - On day $3$, the feeder spits out $123$ pellets; - $\cdots$ If the total number of pellets on some day can be evenly divided among these $26$ cats (that is, each cat gets exactly the same number of pellets), then they will live peacefully. Such a day is called a “Peaceful Mealtime Day”. If it cannot be evenly divided, they will fight over the leftovers. Now, the feeder has been scheduled to run for exactly $2026$ days. Please help the worried Xiao Lan calculate: among these $2026$ days, how many days are “Peaceful Mealtime Days” in total?

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This is a result-only fill-in-the-blank problem. You only need to compute the answer and submit it. The answer is an integer; when submitting, fill in only this integer. Any extra content will result in no score.

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