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This problem trains for: AMC-12, AMC-10, AMC-8, AIME.

If a number has 20 distinct positive divisors, it CANNOT be a multiple of:

Since:

m01

the number may have one of the following prime factorizations:

Any prime raised to a power of: 1, 9, 3 or 4 can be a factor of the number. Therefore:

could be factors.

Also,

could be a factor.

However:

and cannot be a factor.