what is the smallest positive integer that has four different prime factors
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There isn't any number that has four of the same factors. You might be thinking of prime factorizations. In that case, 34 = 81. But while 3 is a factor of 81, 3, 3, 3 and 3 are not.
36
If you mean 2, 3 and 7 as factors then the smallest number is 42
The smallest number that has 9 and 12 as factors is 36.
That number is 724.
4096