Unraveling the Mystery of the Smallest Positive Integer with Exactly 21 Factors

Understanding the intricate world of number theory, particularly the concept of positive factors, can be quite fascinating. Today, we explore the specific problem of identifying the smallest positive integer that has exactly 21 positive factors. This requires a deep dive into the principles of prime factorization and how it influences the number of positive factors a given integer can have.

The Principle of Prime Factorization

Let's begin by recalling a fundamental principle from number theory. If a positive integer n has a prime factorization of the form:

n p_1^{e_1} times p_2^{e_2} times ldots times p_k^{e_k}

then the total number of positive factors of n, denoted as dn, is calculated as:

dn (e_1 1) times (e_2 1) times ldots times (e_k 1)

Determining 21 as the Number of Factors

To have exactly 21 positive factors, we need to express 21 in the form of a product of integers greater than 1. Note that 21 can be factored in several ways:

t21 21 t21 7 times 3 t21 3 times 7

Case Analysis

Case 1: 21 21

This corresponds to a number of the form n p_1^{20}. Using the smallest prime, 2:

n 2^{20} 1,048,576

Case 2: 21 7 times 3

This form suggests a number of the type n p_1^{6} times p_2^{2}. With the smallest primes 2 and 3:

n 2^{6} times 3^{2} 64 times 9 576

Case 3: 21 3 times 7

This configuration implies n p_1^{2} times p_2^{6}. Again, using the smallest primes 2 and 3:

n 2^{2} times 3^{6} 4 times 729 2,916

Conclusion

Comparing the results from these cases:

t2^{20} 1,048,576 t2^{6} times 3^{2} 576 t2^{2} times 3^{6} 2,916

It is clear that the smallest positive integer among these is 576. Therefore, the smallest positive integer that has exactly 21 positive factors is:

boxed{576}

Additional Insights

Given that 21 is an odd number, it suggests that the integer we seek is likely to be a square number. This insight can help in eliminating some options quickly. Starting with 3600, as 60 is a number with many factors, and 3600 is 60 squared, we can count the factors directly. However, we quickly realize that 3600 already has more than 21 factors, so it is clear we need to look for another square number.

Final Thoughts

The beauty of this problem lies in the application of prime factorization and the strategic use of smallest primes. Understanding how different factorizations impact the count of factors can be both enlightening and challenging.