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Positive Divisors Calculator

Reviewed by Calculator Editorial Team

Find all positive divisors of a number with our free positive divisors calculator. Divisors are numbers that divide another number exactly without leaving a remainder. This calculator helps you determine all positive divisors of any integer, which is useful in number theory, cryptography, and various mathematical applications.

What Are Positive Divisors?

A positive divisor of a number is an integer that divides that number exactly without leaving a remainder. For example, the positive divisors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 evenly.

Divisors are fundamental in number theory and have applications in cryptography, computer science, and engineering. Understanding divisors helps in solving problems related to factors, multiples, and prime numbers.

How to Find Divisors

Finding all positive divisors of a number involves checking all integers from 1 up to the number itself to see if they divide the number exactly. Here's a step-by-step method:

  1. Start with the number you want to find divisors for.
  2. Check each integer from 1 to the number to see if it divides the number exactly.
  3. If the number is divisible by the integer, it is a divisor.
  4. Continue this process until you reach the number itself.

Example

Find all positive divisors of 18:

  1. 18 ÷ 1 = 18 → 1 is a divisor.
  2. 18 ÷ 2 = 9 → 2 is a divisor.
  3. 18 ÷ 3 = 6 → 3 is a divisor.
  4. 18 ÷ 4 = 4.5 → 4 is not a divisor.
  5. 18 ÷ 5 = 3.6 → 5 is not a divisor.
  6. 18 ÷ 6 = 3 → 6 is a divisor.
  7. 18 ÷ 7 ≈ 2.57 → 7 is not a divisor.
  8. 18 ÷ 9 = 2 → 9 is a divisor.
  9. 18 ÷ 18 = 1 → 18 is a divisor.

The positive divisors of 18 are: 1, 2, 3, 6, 9, 18.

Divisors Formula

The formula for finding all positive divisors of a number involves checking each integer from 1 to the number itself. There is no direct formula to find divisors, but you can use the following approach:

Divisors Algorithm

  1. For a given number n, initialize an empty list to store divisors.
  2. Iterate through all integers i from 1 to n.
  3. If n is divisible by i (i.e., n % i == 0), add i to the list of divisors.
  4. After the loop, the list will contain all positive divisors of n.

This method ensures that you find all positive divisors of any given number. The time complexity of this algorithm is O(n), which means it becomes less efficient for very large numbers.

Divisors Examples

Here are some examples of finding positive divisors using the calculator:

Example 1: Divisors of 20

Using the calculator, enter 20 and click "Calculate". The result will show the divisors: 1, 2, 4, 5, 10, 20.

Example 2: Divisors of 36

Enter 36 in the calculator and click "Calculate". The result will display the divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Example 3: Divisors of 17

Enter 17 in the calculator and click "Calculate". The result will show the divisors: 1, 17.

Divisors Table

The following table shows the positive divisors for numbers from 1 to 20:

Number Positive Divisors
1 1
2 1, 2
3 1, 3
4 1, 2, 4
5 1, 5
6 1, 2, 3, 6
7 1, 7
8 1, 2, 4, 8
9 1, 3, 9
10 1, 2, 5, 10
11 1, 11
12 1, 2, 3, 4, 6, 12
13 1, 13
14 1, 2, 7, 14
15 1, 3, 5, 15
16 1, 2, 4, 8, 16
17 1, 17
18 1, 2, 3, 6, 9, 18
19 1, 19
20 1, 2, 4, 5, 10, 20

FAQ

What is the difference between divisors and factors?

Divisors and factors are often used interchangeably, but technically, factors are non-negative integers that divide a number exactly, while divisors can include negative integers. However, in most contexts, especially in basic mathematics, the terms are used synonymously.

How do I find the number of divisors of a number?

To find the number of positive divisors of a number, you can use the prime factorization method. First, factorize the number into its prime factors, then add 1 to each of the exponents in the prime factorization, and finally multiply these together. For example, for 12 = 2² × 3¹, the number of divisors is (2+1) × (1+1) = 6.

Can a number have only one positive divisor?

Yes, a number can have only one positive divisor if it is a prime number. Prime numbers are only divisible by 1 and themselves, so their only positive divisor is 1 and the number itself.

What are the properties of divisors?

Divisors have several important properties, including:

  • Every number has at least two divisors: 1 and itself.
  • Divisors are always integers.
  • The number of divisors of a number is related to its prime factorization.
  • Divisors can be used to find the greatest common divisor (GCD) and least common multiple (LCM) of numbers.