Let's work backwards to find the minimum number of boxes required to pack all the erasers.
First, we need to find the total number of erasers. Since the erasers are sorted equally into 42 containers, we can represent the number of erasers in each container with the variable x. Thus, the total number of erasers is 42x.
Since we need to repack the erasers into boxes of a dozen, we need to divide 42x by 12. However, this division must result in a non-zero perfect cube.
Let's start dividing by 12 and see when we get a perfect cube:
- When x = 1, 42x/12 = 3.5, which is not a perfect cube.
- When x = 2, 42x/12 = 7, which is not a perfect cube.
- When x = 3, 42x/12 = 10.5, which is not a perfect cube.
- When x = 4, 42x/12 = 14, which is not a perfect cube.
- When x = 5, 42x/12 = 17.5, which is not a perfect cube.
- When x = 6, 42x/12 = 21, which is not a perfect cube.
- When x = 7, 42x/12 = 24.5, which is not a perfect cube.
- When x = 8, 42x/12 = 28, which is not a perfect cube.
- When x = 9, 42x/12 = 31.5, which is not a perfect cube.
- When x = 10, 42x/12 = 35, which is not a perfect cube.
We haven't found a perfect cube yet, so let's try a different approach. We know that 42x must be divisible by 12 to get a whole number of boxes, so let's factor out 12 from 42x:
42x = 2 x 3 x 7 x x
Since 42x is a perfect cube, we know that x must have factors of 2, 3, and 7 to make the product a perfect cube. Let's try different values of x until we find the smallest one that works:
- When x = 2 x 3 x 7 = 42, 42x/12 = 147, which is a perfect cube (7^3).
- When x = 2 x 3 x 7 x 2 = 84, 42x/12 = 294, which is a perfect cube (6^3).
- When x = 2 x 3 x 7 x 3 = 126, 42x/12 = 441, which is a perfect cube (9^3).
Thus, the minimum number of boxes required to pack all the erasers is 147.
A stationery distributor sorted out his erasers equally into 42 containers. The erasers in each container are all repacked into boxes of a dozen erasers each. Knowing that the original total number of erasers is a non-zero perfect cube, what is the minimum number of boxes required to pack all the erasers.
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