Integer arithmetic computes with integers. Addition, subtraction and multiplication preserve integer values; division can instead return a quotient and remainder through Euclidean division. For example, gives five complete groups and a remainder of two. The ceiling count for fixed-size batches requires six groups to hold all 102 items.
Mathematical integers have no fixed size bound. A computer representation may have one, so an exact arithmetic formula still requires sufficient space for its inputs and intermediate values. The floor function and ceiling function define mathematical bounds independently of a chosen representation.
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