What is the smallest positive integer with exactly 12 (positive) divisors?
36 (2^2 3^2) seems likely
1 2 3 4 6 9 12 18 36 nope
48 (2^4 3^1)?
1 2 3 4 6 8 12 16 24 48 nope
72 (2^3 3^2)?
1 2 3 4 6 8 9 12 18 24 36 72
Yes
To find the smallest positive integer with exactly 12 positive divisors, we need to understand the concept of divisors.
A divisor of a positive integer is a number that divides evenly into that integer without leaving a remainder. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12.
To determine the number of divisors a positive integer has, we can use the prime factorization of that integer. The prime factorization expresses the integer as a product of prime numbers.
In general, if the prime factorization of a positive integer is given as p1^a1 * p2^a2 * p3^a3 * ... * pn^an, where p1, p2, ..., pn are prime numbers and a1, a2, ..., an are positive integers, then the total number of divisors can be found using the formula (a1 + 1) * (a2 + 1) * (a3 + 1) * ... * (an + 1).
In the case of the smallest positive integer with exactly 12 divisors, we are looking for an integer that has a prime factorization of the form p1^a1 * p2^a2, where p1 and p2 are distinct prime numbers, and a1 and a2 are positive integers.
We can start by considering the prime factorization with the smallest possible values for a1 and a2, which are both 1. In this case, the prime factorization is p1 * p2, and by the earlier formula, the total number of divisors is (1 + 1) * (1 + 1) = 4.
Since we need exactly 12 divisors, we need to find a prime factorization with a higher number of divisors. The next option is to set one of the exponents, a1 or a2, to 2. Considering a1 = 2, the prime factorization becomes p1^2 * p2, and the total number of divisors is (2 + 1) * (1 + 1) = 6.
This is still not the desired 12 divisors, so we continue exploring the possibilities. The next option is to set the other exponent, a2, to 2. With a2 = 2, the prime factorization becomes p1 * p2^2, and the total number of divisors is (1 + 1) * (2 + 1) = 6.
Again, this falls short of the required 12 divisors. Finally, we can try setting both exponents to 2, which results in p1^2 * p2^2. The number of divisors is (2 + 1) * (2 + 1) = 9.
Since 9 is not equal to 12, we continue considering larger exponents. By setting a1 = 3 and a2 = 1, the prime factorization becomes p1^3 * p2, resulting in (3 + 1) * (1 + 1) = 8 divisors.
Similarly, with a1 = 1 and a2 = 3, the prime factorization becomes p1 * p2^3, resulting in (1 + 1) * (3 + 1) = 8 divisors.
None of these options satisfy the required 12 divisors. However, if we set both exponents to 3, we get p1^3 * p2^3, resulting in (3 + 1) * (3 + 1) = 16 divisors.
Thus, the smallest positive integer with exactly 12 divisors cannot be obtained by using prime factorizations with exponents less than or equal to 3.
To find the smallest positive integer with exactly 12 divisors, we need to consider prime factorizations with exponents greater than 3. By using higher values for a1 and a2, we could get a number of divisors that is larger than 12, so it is not necessary to continue exploring those possibilities.
Hence, the smallest positive integer with exactly 12 divisors is obtained by setting a1 = 2 and a2 = 3, resulting in the prime factorization p1^2 * p2^3, and the total number of divisors is (2 + 1) * (3 + 1) = 12.
Since we want the smallest positive integer, we can assign the smallest distinct prime numbers to p1 and p2, which are 2 and 3:
2^2 * 3^3 = 4 * 27 = 108
Therefore, the smallest positive integer with exactly 12 divisors is 108.