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What is the smallest number with exactly 14 divisors?

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Weekly Problem 18 - 2010

Stage: 3 Short Challenge Level: Challenge Level:1

The solution depends on how the cube is painted.

If you paint the large cube so that two of the small corner cubes have their three painted faces all painted the same colour, then $12$ of the small cubes will have at least one red face and also at least one blue face.

If you paint the large cube so that no small corner cube has all three of its painted faces painted the same colour, then $16$ small cubes will have at least one red face and one blue face.

This problem is taken from the UKMT Mathematical Challenges.

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