Problem

A \(10\text{-}by-9\text{-}by-8\text{-}\text{inch}\) container receives \(2\text{-}by-3\text{-}by-4\text{-}\text{inch}\) boxes in the corresponding orientation shown. Calculate the whole-box count and remainder along each axis, state whether the boxes fit, and find the total count.

An aligned rectangular container and sample box are dimensioned 10 by 9 by 8 inches and 2 by 3 by 4 inches in matching orientation; counts and fit decision are withheld.

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Respect the stated orientation by pairing each container dimension with its corresponding item dimension. Along each axis, keep only whole copies using floor division and record any leftover length; a fit requires at least one whole copy in every direction. Multiply the three independent axis counts to obtain the aligned total, rather than comparing volumes alone.

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Using only the first two axes gives floor(10/2)floor(9/3)=5(3)=15 boxes, with the 8-inch height and 4-inch box dimension omitted.

Reordering each box as 4 by 3 by 2 gives counts floor(10/4)=2, floor(9/3)=3, and floor(8/2)=4, for 2(3)(4)=24 boxes in a different orientation.

The axis counts are floor(10/2)=5, floor(9/3)=3, and floor(8/4)=2, with remainders 0, 0, and 0. The boxes fit in the stated orientation, and 5(3)(2)=30 boxes fit.

The axis counts are 5, 3, and 2 with remainders 0, 0, and 0, but adding the counts gives 5+3+2=10 boxes.

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A \(10\text{-}by-9\text{-}by-8\text{-}\text{inch}\) container receives \(2\text{-}by-3\text{-}by-4\text{-}\text{inch}\) boxes in the corresponding orientation shown. Calculate the whole-box count and remainder along each axis, state whether the boxes fit, and find the total count.

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