Question: A deep learning model processes patient data in batches, each batch size a multiple of 11, and the total dataset size is a multiple of 9. If the model must process at least 100 but fewer than 200 patients, what is the smallest possible batch size (in patients) that allows full utilization of both constraints? - RoadRUNNER Motorcycle Touring & Travel Magazine

Understanding the Context

When deploying deep learning models for patient data processing, choosing an optimal batch size is critical for performance, memory usage, and training speed. In a recent scenario, a healthcare AI system must process patient records in full batches, where each batch size is a multiple of 11, and the total dataset size is a multiple of 9. With the dataset limited to between 100 and 199 patients, researchers seek the smallest valid batch size that satisfies both constraints—ensuring efficient computation and full data utilization.

Understanding the Constraints

• Batch size must be a multiple of 11.
  So possible batch sizes include: 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187

Key Insights

• Total dataset size is a multiple of 9, and lies between 100 and 199.
  Multiples of 9 in this range:
  108, 117, 126, 135, 144, 153, 162, 171, 180, 189

• Each batch must fit completely—i.e., the batch size must divide the total dataset size evenly.
  

• We want the smallest batch size (multiple of 11) such that some multiple of it is a multiple of 9 within the 100–199 range.

But since the total dataset size must itself be a multiple of 9, we look for the smallest multiple of 11 that divides at least one number in {108, 117, ..., 189}. In other words, find the smallest b = 11×k such that b divides one of the multiples of 9 between 100 and 199.

Searching for the Smallest Valid Batch Size

Final Thoughts

We test successive multiples of 11, starting from the smallest, and check if any are divisors of a multiple of 9 in the 100–199 range.

1. Batch size = 11
   Check if any multiple of 9 between 100–199 is divisible by 11:
   

   • 108 ÷ 11 ≈ 9.82 → No
     
   • 117 ÷ 11 ≈ 10.63 → No
     
   • 126 ÷ 11 ≈ 11.45 → No
     
   • 135 ÷ 11 ≈ 12.27 → No
     
   • 144 ÷ 11 ≈ 13.09 → No
     
   • 153 ÷ 11 ≈ 13.90 → No
     
   • 162 ÷ 11 ≈ 14.73 → No
     
   • 171 ÷ 11 ≈ 15.55 → No
     
   • 180 ÷ 11 ≈ 16.36 → No
     
   • 189 ÷ 11 ≈ 17.18 → No
     → 11 does not divide any multiple of 9 in 100–199

2. Batch size = 22
   

   • 108 ÷ 22 ≈ 4.91 → No
     
   • 117 ÷ 22 ≈ 5.32 → No
     
   • 126 ÷ 22 ≈ 5.72 → No
     
   • 135 ÷ 22 ≈ 6.14 → No
     → No match

3. Batch size = 33
   33 is a multiple of 11 (11×3). Try dividing multiples of 9:
   

   • 108 ÷ 33 ≈ 3.27 → No
     
   • 117 ÷ 33 ≈ 3.545 → No
     → No

4. Batch size = 44
   

   • 108 ÷ 44 ≈ 2.45 → No
     
   • 144 ÷ 44 ≈ 3.27 → No
     → No

5. Batch size = 55
   

   • 110 ≤ 108–199 → 108 ÷ 55 ≈ 1.96, 117 ÷ 55 ≈ 2.12 → No
     → No