Rainbow Building

Number Theory Level 4

You are building a rainbow building from \(N\) cubic unit blocks.

First, you put 3 cubes each to form a group of columns and discard the remaining cubes away.

Then, you put 5 columns together each to form a group of \(3\times 5\) cubic bases and discard the remaining columns away as before.

Finally, you stack 7 bases over one another to build a desired \(3\times 5\times 7\) cuboid structure each, as shown above, and discard all the other bases away as usual.

If there are a total of 52 discarded cubes, and \(N\) is a multiple of 11, what is the least possible value of \(N\)?


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