Divisor Calculator: All Divisors, Count, Sum and Factorization
Enter any positive integer to see every number that divides it evenly. The calculator lists all divisors in ascending order, counts them, adds them up, breaks down the prime factorization, and identifies proper divisors. Enter a second number to find common divisors and the greatest common divisor (GCD). All results update instantly as you type.
What is a divisor?
A divisor of a positive integer n is any positive integer that divides n exactly, leaving no remainder. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12 - each divides 12 with a quotient that is also a whole number. The terms "divisor" and "factor" are interchangeable in this context. Every positive integer has at least two divisors: 1 and itself. The only exception is the number 1, which has just one divisor (itself), because 1 divided by 1 is 1.
How the calculator finds all divisors
The efficient way to find every divisor of n is to test only the integers from 1 up to the square root of n. For each integer i that divides n evenly, both i and n/i are divisors, so each test yields a pair. For example, when n = 60 and i = 4, we get the pair (4, 15) because 60 / 4 = 15. Testing up to sqrt(60) ~ 7.7 finds all divisors without checking every number up to 60. The result is then sorted into ascending order. The number of divisors can also be computed directly from the prime factorization: if n = p1^a1 x p2^a2 x ... x pk^ak, then the total count of divisors is (a1+1)(a2+1)...(ak+1).
Prime factorization and the divisor count formula
Every integer greater than 1 can be written as a unique product of prime powers (the Fundamental Theorem of Arithmetic). For instance, 60 = 2^2 x 3 x 5. From this representation, the number of divisors is the product of each exponent plus one: (2+1)(1+1)(1+1) = 3 x 2 x 2 = 12. You can verify this against the full list: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The sum of all divisors (sigma function) follows a similar formula using geometric series: for each prime power p^a, the contribution is (p^(a+1) - 1) / (p - 1). For 60, the sum is (1+2+4)(1+3)(1+5) = 7 x 4 x 6 = 168.
Proper divisors, perfect numbers, and number classification
A proper divisor of n is any divisor except n itself. The number 6 has proper divisors 1, 2, and 3, which add up to exactly 6 - making 6 a perfect number. Numbers where the proper-divisor sum is less than the number are called deficient (most numbers fall here); those where it exceeds the number are abundant (12 is the smallest: 1+2+3+4+6 = 16 > 12). These classifications come from ancient Greek mathematics and still appear in modern number theory. For two numbers, the greatest common divisor (GCD) is the largest value in their common divisor list, and the common divisors are exactly the divisors of the GCD itself.
Divisor classification guide
| Type | Condition | Examples |
|---|---|---|
| Prime | Exactly 2 divisors (1 and itself) | 2, 3, 5, 7, 11, 13, 17, 19, 23 |
| Composite | More than 2 divisors | 4, 6, 8, 9, 10, 12, 14, 15 |
| Perfect | Sum of proper divisors = itself | 6, 28, 496, 8128 |
| Deficient | Sum of proper divisors < itself | 1, 2, 3, 4, 5, 7, 8, 9, 10 |
| Abundant | Sum of proper divisors > itself | 12, 18, 20, 24, 36, 40, 48 |
| Highly composite | More divisors than all smaller positive integers | 1, 2, 4, 6, 12, 24, 36, 48, 60 |
How numbers are classified based on their divisors and the sum of proper divisors.
Frequently asked questions
What is the difference between a divisor and a factor?
In everyday arithmetic the words are interchangeable. Both refer to a positive integer that divides a given number without a remainder. "Factor" is more common in multiplication contexts (3 and 4 are factors of 12), while "divisor" is used in division contexts (12 divided by 3 has no remainder, so 3 is a divisor). This calculator uses both terms to mean the same thing.
How do I find all divisors of a large number quickly?
The most efficient manual method is to find the prime factorization first, then generate divisors by combining all possible prime-power combinations. For the calculator, the algorithm tests only integers up to the square root of the number, which cuts the work roughly in half. For numbers up to 1,000,000, this is nearly instant in a browser.
What are proper divisors?
Proper divisors of a number n are all divisors of n except n itself. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6. The concept matters in number classification: if the sum of proper divisors equals n, the number is perfect; if less, it is deficient; if more, it is abundant.
How is the GCD related to common divisors?
The common divisors of two numbers are exactly the divisors of their greatest common divisor (GCD). For example, gcd(60, 48) = 12, and the divisors of 12 are 1, 2, 3, 4, 6, and 12 - which are also all the numbers that divide both 60 and 48 evenly. This is why finding the GCD first is an efficient shortcut.
Can 1 be a divisor?
Yes. The number 1 divides every positive integer exactly (n / 1 = n with remainder 0), so 1 is always the smallest divisor of any positive integer. Similarly, n always divides itself, making n its own largest divisor. These two are called the trivial divisors.
What does it mean for a number to be highly composite?
A highly composite number has more divisors than any smaller positive integer. The sequence begins 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... The number 60 has 12 divisors, more than any positive integer below it, which is why 60 minutes and 60 seconds have been used as base units for time: 60 can be divided evenly by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.