192, although there are smaller numbers with more.
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192
The smallest is 192 but there are infinitely many such numbers. For example, take any prime number to the 13th power and the result will have exactly fourteen factors. The smallest is 192.
The smallest prime factors are 2,3,5,7 thus the smallest number would be the product of these will be 210
240 is the smallest number with twenty factors.
48 is.