But the condition is not that $ B $ divides $ D $, but that $ D $ is a multiple of 9 and $ B $ divides $ D $ â so $ B $ must be a divisor of some multiple of 9.

February 22, 2026

But the condition is not that $ B $ divides $ D $, but that $ D $ is a multiple of 9 and $ B $ divides $ D $ â so $ B $ must be a divisor of some multiple of 9.

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Therefore, the smallest possible batch size that allows full utilization (i.e., divides a valid dataset size) is $ oxed{198} $.
But wait: the model can use any multiple of 9, not necessarily including 11. The batch size $ B $ must be such that some valid dataset (divisible by 9, â¥100, <200) can be divided into batches of size $ B $. Since $ B $ must divide the dataset size, $ B $ must divide some multiple of 9 in [100,199].
But any integer $ B $ divides some multiple of 9 (e.g., take $ D = B \cdot 9 $, but again, $ B $ must be divisible by 11).
But every positive integer divides some multiple of 9 (e.g., $ D = B \cdot 9 $ only if $ B \mid D $, so if we set $ D = B \cdot k $, we need $ 9 \mid Bk $. But we can choose $ k $).
To allow $ B $ to divide a number $ D \in [100,199] \cap 9\mathbb{Z} $, it suffices that $ B $ divides at least one such $ D $.
So we seek the smallest multiple of 11 that divides at least one multiple of 9 in [100,199].

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