To determine the smallest whole number with exactly 24 different factors, we need to understand the relationship between a number's prime factorization and its total number of divisors.

Formula for the Number of Divisors

For a number $n$ with prime factorization:

$n = p_1^{e_1} \cdot p_2^{e_2} \cdot \dots \cdot p_k^{e_k}$

where $p_1, p_2, \dots, p_k$ are distinct primes, the total number of divisors of $n$ is given by the formula:

$D(n) = (e_1 + 1)(e_2 + 1) \dots (e_k + 1)$

where $e_1, e_2, \dots, e_k$ are the exponents of the primes in the factorization of $n$.

We are looking for the smallest $n$ such that $D(n) = 24$.

Step 1: Factorizing 24

The number of divisors, 24, can be factorized into different combinations of factors. We can break it down into the following possibilities:

$24 = 2^3 \times 3$

This implies several combinations of the exponents $e_1 + 1, e_2 + 1, \dots$, where we multiply the factors to get 24. We will explore these combinations to find the smallest number.

Step 2: Try Different Factor Combinations

1. $24 = 2^3 \times 3$ implies the following factorization of the exponents:

   • $(e_1 + 1) = 8$, which gives $e_1 = 7$
   • $(e_2 + 1) = 3$, which gives $e_2 = 2$
   • $(e_3 + 1) = 1$, which gives $e_3 = 0$

   So, the number is $p_1^7 \times p_2^2$. The smallest such number occurs when $p_1 = 2$ and $p_2 = 3$, so the number is:

   $n = 2^7 \times 3^2 = 128 \times 9 = 1152$

2. Alternative factorization: Another approach would be using $24 = 4 \times 6 \times 1$, which would give:

   • $(e_1 + 1) = 4$, so $e_1 = 3$
   • $(e_2 + 1) = 6$, so $e_2 = 5$

   So, the number is $p_1^3 \times p_2^5$. Using $p_1 = 2$ and $p_2 = 3$, we get:

   $n = 2^3 \times 3^5 = 8 \times 243 = 1944$

Step 3: Conclusion

The smallest number with exactly 24 divisors is 1152.

Would you like more details or have any questions about how to approach this problem? Here are some related questions to expand your understanding:

1. How can we calculate the number of divisors for a given number?
2. What happens if we increase the number of factors beyond 24? How does it affect the size of the number?
3. Can we find other combinations for the factorization of 24 to get different results?
4. How can we apply this method to find the smallest number with any given number of divisors?
5. What is the relationship between prime factorization and divisor count in general?

Tip: When trying to find a number with a given number of divisors, it's useful to start with the smallest primes and try various factorizations of the number of divisors.