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Q: What is the smallest number that has 4 different prime factors?

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How about: 105

1155

30,030 Factorization: 2x3x5x7x11x13

210 with factors of 2, 3, 5, and 7

The smallest number with three different prime factors is 30.

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Logically, the smallest number with 4 different prime factors would have to be the product of the smallest four prime numbers. So the smallest natural number with four different prime factors is 210 (2*3*5*7).

How about: 105

1155

As a product of its prime factors: 2*3*5*7 = 210

5 and 11

30,030 Factorization: 2x3x5x7x11x13

1155

what is the smallest positive integer that has four different prime factors

If a number has 5 factors then it most certainly is not prime. Hence there is some confusion as to what you are asking here.

210 with factors of 2, 3, 5, and 7

210, welcome :D

154

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