Counting Positive Divisors Less Than Square Root of a Positive Integer

This article provides a step-by-step guide on how to count the number of positive divisors of a positive integer n that are less than the square root of n. We will cover the prime factorization method, the formula for total divisors, and how to apply these techniques to calculate the desired count.

Prime Factorization and Total Divisors

The first step in counting the positive divisors of a positive integer n is to determine its prime factorization. Suppose n can be written as:

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

where p_1, p_2, ldots, p_k are distinct prime factors of n, and e_1, e_2, ldots, e_k are their respective positive integer exponents.

The total number of positive divisors of n is given by the formula:

d_n (e_1 1)(e_2 1) ldots (e_k 1)

Divisors and the Square Root

Next, we need to break down the divisors into pairs. If d is a divisor of n, then frac{n}{d} is also a divisor. These pairs include one divisor less than or equal to the square root of n and one that is greater than or equal to the square root of n.

To find the number of divisors less than the square root of n, we can use the following formula:

- If n is a perfect square, the number of divisors less than sqrt{n} is frac{d_n - 1}{2}.

- Otherwise, if n is not a perfect square, the number of divisors less than sqrt{n} is frac{d_n}{2}.

Example Calculation

Let's illustrate this method with an example. Consider n 36.

Prime Factorization

The prime factorization of 36:

36 2^2 times 3^2

Counting the total divisors using the formula:

d_{36} (2 1)(2 1) 3 times 3 9

The divisors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Identifying Perfect Square

Since 36 is a perfect square, sqrt{36} 6.

Counting Divisors Less Than 6

The divisors less than 6 are: 1, 2, 3, 4, totaling 4.

Using the formula for non-perfect squares:

text{Divisors less than } sqrt{36} frac{d_{36} - 1}{2} frac{9 - 1}{2} 4

Thus, the method accurately gives us the count of positive divisors of 36 that are less than its square root.

Conclusion

The process described here for counting the positive divisors of a positive integer less than its square root can be applied to any positive integer. By understanding the prime factorization, calculating the total divisors, and applying the divisor counting formula, one can efficiently determine this important number theoretic property.

Keywords: positive divisors, square root, prime factorization