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230

Q: What is the prime factorization of the smallest positive integer that has exactly 31 factors?

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Factorization into its smallest factors.

Among positive integers, 6

30,030 Factorization: 2x3x5x7x11x13

4096 is the smallest number with exactly 13 factors. (Including 1 and itself as factors).192 is the smallest number with exactly 13 properfactors, (excluding the number itself).120 is the smallest number with at least 13 factors (it has 16 factors)To find the smallest number with exactly N factors (for example, N=12):Factor N into primes, (for example, 12= 2*2*3Subtract 1 from each of the prime factors (for example, 1,1,2)Sort them by largest first (for example, 2,1,1)Raise successive prime numbers to these powers (for example, 2^2,3^1,5^1)Multiply these together (for example: 4*3*5 = 60; 60 is the smallest number with exactly 12 factors).

6

Related questions

230

Factorization into its smallest factors.

Among positive integers, 6

3

4 and 9

The smallest number that has exactly eight factors is 24. The factors of 1,2,3,4,6,8,12, and 24

The positive integer factors of 293 are: 1, 293

30,030 Factorization: 2x3x5x7x11x13

30

The smallest number with exactly eight factors is 24.

4096 is the smallest number with exactly 13 factors. (Including 1 and itself as factors).192 is the smallest number with exactly 13 properfactors, (excluding the number itself).120 is the smallest number with at least 13 factors (it has 16 factors)To find the smallest number with exactly N factors (for example, N=12):Factor N into primes, (for example, 12= 2*2*3Subtract 1 from each of the prime factors (for example, 1,1,2)Sort them by largest first (for example, 2,1,1)Raise successive prime numbers to these powers (for example, 2^2,3^1,5^1)Multiply these together (for example: 4*3*5 = 60; 60 is the smallest number with exactly 12 factors).

192