There are four numbers under 100 that have exactly 12 factors.
To determine which numbers under 100 have 12 factors, we need to consider the possible prime factorizations that would result in 12 factors. The number of factors of a number can be found by adding 1 to each exponent in its prime factorization and then multiplying these results. Thus, to get 12 factors, we need to find combinations of exponents that multiply to 12 when each is incremented by one. Here are the possibilities:
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12 = 12. This corresponds to a prime factorization of the form p11, where p is a prime number. Since 211 is already greater than 100, this case yields no solutions less than 100.
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12 = 6 2. This corresponds to a prime factorization of the form p5 q1, where p and q are distinct prime numbers.
- If p = 2, we have 25 q = 32 q. If q = 3, we have 32 3 = 96, which has (5+1)(1+1) = 62 = 12 factors.
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12 = 4 3. This corresponds to a prime factorization of the form p3 q2, where p and q are distinct prime numbers.
- If p = 2, we have 23 q2 = 8 q2. If q = 3, we have 8 9 = 72, which has (3+1)(2+1) = 43 = 12 factors.
- If p = 3, we have 33 q2 = 27 q2. There are no solutions here as with q = 2, the product is > 100.
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12 = 3 2 2. This corresponds to a prime factorization of the form p2 q1 r1, where p, q, and r are distinct prime numbers.
- If p = 2, q = 3, r = 5, we have 22 3 5 = 4 3 5 = 60, which has (2+1)(1+1)(1+1) = 322 = 12 factors.
- If p = 2, q = 3, r = 7, we have 22 3 7 = 4 3 7 = 84, which has (2+1)(1+1)(1+1) = 322 = 12 factors.
Therefore, the numbers under 100 with 12 factors are 60, 72, 84, and 96.
