What is the least positive integer that has exactly thirteen

factors?

To find the least positive integer that has exactly thirteen factors, we need to understand the concept of factors.

Factors are the numbers that multiply to give a certain number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because these numbers multiply together to give 12.

To determine the number of factors a given number has, we need to find its prime factorization. The prime factorization is the representation of a number as a product of its prime factors.

Now, let's find the least positive integer with exactly thirteen factors:

1. Start by writing out the prime numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

2. Next, express the number of factors as a product of prime numbers. Since we want exactly thirteen factors, we need to find numbers that can be raised to certain powers to result in the product 13.

We can have:
- One prime number raised to the 12th power: p^12
- One prime number raised to the 6th power, and another prime number raised to the power of 2: p^6 * q^2, where p and q are distinct prime numbers.

3. Now, look for a combination that gives us the least positive integer. We start with the first case:

- One prime number raised to the 12th power: p^12
The least prime number we can use is 2. Therefore, our first case becomes 2^12.

- One prime number raised to the 6th power, and another prime number raised to the power of 2: p^6 * q^2
The least prime numbers we can choose are 2 and 3. Therefore, our second case becomes 2^6 * 3^2.

4. Compare the values we obtained from the two cases:

- 2^12 = 4096
- 2^6 * 3^2 = 288

Since we are searching for the least positive integer, our answer is 288. Hence, the least positive integer with exactly thirteen factors is 288.