What is the smallest number with 10 different factors( other than 1)?

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justaguide eNotes educator| Certified Educator

This problem is a little tricky. To find the smallest number that has 10 factors, we need one that can be created by multiplying the smallest number of prime numbers which are small too.

If a number has only 2, it has 1 factor. With 2*2, we have 2 factors, with 2*3 we have three factors 2, 3, 6. With 2*2*3, we have 5 factors 2, 3, 4, 6, 12. With 2*2*3*3 we have 8 factors 2, 3, 4, 6, 9, 12, 18, 36. With 2*2*2*3*3 we get 10 factors.

This gives the smallest number with 10 factors as 8*9 = 72. We can verify that 72 has 10 factors which are: 2, 3, 4, 6, 8, 9, 12, 18, 24, 72

The required smallest number with 10 different factors is 72.

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