The following challenge appeared in Mathematical Pie, a Mathematical Association publication.

The prime factorisation of 600 is 2^{3} x 5^{2} x 3. We can make all the factors of 600 by choosing from four possibilities for the 2 (to include it 0, 1, 2 or 3 times); three possibilities for the 5 (to include it 0, I or 2 times) and two possibilities for the 3 (to include it or not).

Altogether 4 x 3 x 2 = 24 possibilities (if we don't choose any of the three this will give the factor l). So 600 has 24 factors.

Can you use the same idea to find all the numbers below 1000 which have exactly 20 factors?

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