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Home » #### Sum: \(\frac72\), Product: \(\frac32\) - Appfinity Technologies

#### Sum: \(\frac72\), Product: \(\frac32\) - Appfinity Technologies

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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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 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.

📌 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].

📌 List multiples of 9 in range: 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198

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