Therefore, the least common multiple of the quantities the boxes can hold is \(\boxed{36}\).

["Understanding the Least Common Multiple: Why It Matters and Its Value is (\boxed{36})", "When working with quantities that represent box capacities—such as packages, storage units, or batch measurements—finding a common measure that works for all of them is essential. One critical concept in this context is the least common multiple (LCM). It serves as the smallest value that both quantities can divide into evenly, making it invaluable in scheduling, resource allocation, and uniform distribution.", "What Is the Least Common Multiple?\nThe LCM of two or more numbers is the smallest number that is a multiple of each of them. This concept is especially useful when managing systems where different components must align—like coordinating boxes of varying sizes to pack items efficiently. For example, if one box holds 4 items and another holds 9, the smallest number that both fit evenly into multiples of is 36. This is why, therefore, the least common multiple of the relevant quantities is (\boxed{36}).", "Why LCM Matters in Quantity Problems\n- Synchronizing intervals: Whether scheduling deliveries or production batches, the LCM finds the earliest time point where all cycles align.\n- Standardizing units: It helps convert or relate different scales into a shared framework.\n- Minimizing waste: By using the smallest common quantity, resource use becomes optimized.", "In this case, when considering box capacities such that 36 is the smallest number divisible by their respective sizes, it elegantly resolves compatibility issues. Thus, the LCM determines (\boxed{36}) as the optimal value.", "Practical Example\nSuppose two boxes hold 4 and 9 units. To pack items without cuts or gaps:\n- The LCM of 4 and 9 is 36.\n- This means 36 is the smallest batch size where both box types can hold complete sets evenly.\n- Hence, (\boxed{36}) emerges as the least common multiple governing efficient packing.", "Conclusion\nThe least common multiple is more than a math concept—it’s a real-world tool that simplifies complex systems. With a value of (\boxed{36}), it provides clarity, uniformity, and efficiency in handling box capacities and beyond. Understanding LCM empowers better planning and smoother operations in logistics, manufacturing, education, and daily tasks alike."]