Find the Divisors of Any Integer
Enter an integer to get its divisors, proper divisors, prime factorization, and useful number properties.
What Is a Divisor?
A divisor (also called a factor) of an integer n is any integer that divides n with no remainder. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12.
Divisors are foundational in arithmetic, number theory, and many practical algorithms. If you are studying prime numbers, greatest common divisor (GCD), least common multiple (LCM), modular arithmetic, or cryptography, divisor analysis is one of the first tools you should master.
Quick Examples
- 18: divisors are 1, 2, 3, 6, 9, 18
- 25: divisors are 1, 5, 25
- 29: divisors are 1, 29 (so 29 is prime)
How This Divisors Calculator Works
This calculator checks integers from 1 up to the square root of your number. Whenever it finds a divisor d, it also finds the paired divisor n/d. This makes the process much faster than testing all numbers up to n.
What You Get
- Full list of positive divisors
- Proper divisors (all divisors except the number itself)
- Total number of divisors
- Sum of divisors and sum of proper divisors
- Prime factorization
- Classification: prime/composite and perfect/abundant/deficient
Why Divisors Matter
1) Prime Testing
A number greater than 1 is prime if it has exactly two positive divisors: 1 and itself. Divisor counts instantly reveal whether a number is prime or composite.
2) Perfect, Abundant, and Deficient Numbers
Compare a number with the sum of its proper divisors:
- Perfect: sum equals the number (example: 6)
- Abundant: sum is greater than the number (example: 12)
- Deficient: sum is less than the number (example: 10)
3) Real-World Applications
- Cryptography and modular arithmetic
- Scheduling and cycle analysis
- Hashing and algorithm optimization
- Math education and contest preparation
Frequently Asked Questions
Are negative divisors included?
The calculator lists positive divisors. For a negative number, the positive divisors of its absolute value still apply, and each has a negative counterpart.
What about zero?
Zero is a special case. It has infinitely many divisors because any non-zero integer divides 0.
Can I use decimals?
This tool is for integers only. If you enter a decimal, you will be asked to provide a whole number.
Try These Practice Inputs
- 36 (many divisors)
- 97 (prime number)
- 496 (perfect number)
- 840 (highly composite style example)
If you are learning number theory, run a few values through the calculator and look for patterns in divisor count and prime factorization. That habit builds deep intuition quickly.