Divisor Calculator

What Is a Divisor?

A divisor of an integer is another integer that divides it with no remainder. If d divides n exactly, mathematicians write d ∣ n.

For example, 3 is a divisor of 12 because 12 ÷ 3 = 4, which is an integer. The number 5 is not a divisor of 12 because 12 ÷ 5 leaves a remainder.

Every positive integer has at least two familiar divisors: 1 and the number itself. The only exception to that wording is 1, whose two values are the same. This small detail separates prime numbers from every other positive integer greater than 1.

Why Divisors Matter for Prime Numbers

Prime numbers are defined by their divisor structure. A positive integer greater than 1 is prime when its only positive divisors are 1 and itself. A number with more than two positive divisors is composite.

The calculator therefore does more than list factors. Its result explains why a number is prime or composite. Entering 97 returns only 1 and 97, while entering 99 returns 1, 3, 9, 11, 33, and 99.

Positive, Negative, and Proper Divisors

The word divisor can refer to several related sets. A calculator should state which set it uses rather than mixing them together.

Positive divisors

These are the positive integers that divide a number exactly. The positive divisors of 18 are 1, 2, 3, 6, 9, and 18.

Negative divisors

Each positive divisor has a negative counterpart. The negative divisors of 18 are −1, −2, −3, −6, −9, and −18. Both signs divide 18 without a remainder.

Proper divisors

Proper divisors are the positive divisors excluding the number itself. The proper divisors of 18 are 1, 2, 3, 6, and 9. Their sum helps classify perfect, abundant, and deficient numbers.

Factor Pairs and the Square Root Rule

Divisors arrive in pairs. If 4 divides 36, the matching divisor is 9 because 4 × 9 = 36. The positive factor pairs of 36 are:

• 1 × 36
• 2 × 18
• 3 × 12
• 4 × 9
• 6 × 6

This pairing explains why a divisor search only needs to test integers up to the square root of the number. Once a smaller divisor is found, its larger partner follows from division.

For 36, no new pair begins after 6 because √36 = 6. Testing 9 again would only reproduce the pair 4 × 9 in reverse order.

Why perfect squares have an odd divisor count

Most divisors pair with a different value. A perfect square has one middle pair in which both values are equal. For 36, that pair is 6 × 6. Since 6 is counted once, the total number of positive divisors is odd.

The reverse is also true: a positive integer has an odd number of positive divisors exactly when it is a perfect square.

Prime Factorization Behind the Divisor List

Prime factorization gives a compact description of every divisor a number can have. Consider 360:

Any positive divisor of 360 can use the prime 2 zero to three times, the prime 3 zero to two times, and the prime 5 zero or one time. A divisor therefore has the form:

Choosing exponent 0 means that prime is absent from the divisor. For example, 45 = 2⁰ × 3² × 5¹, so 45 divides 360.

This connection is especially useful for prime numbers. A prime has factorization p¹, so its divisors come only from exponents 0 and 1: 1 and p.

How the Number of Divisors Is Calculated

Suppose the prime factorization of a positive integer is:

For each prime, the exponent in a divisor may range from 0 through the exponent in n. That gives a + 1 choices for each prime. Multiplying the choices gives the positive divisor count:

Example: divisor count of 360

Since 360 = 2³ × 3² × 5¹:

So 360 has 24 positive divisors. When both positive and negative divisors are counted, every positive divisor has one negative partner, giving 48 nonzero integer divisors.

How the Sum of Divisors Is Calculated

The divisor sum function, written as σ(n), adds all positive divisors of n. The sum can be read directly from the list or calculated from the prime factorization.

Example: divisor sum of 12

The positive divisors of 12 are 1, 2, 3, 4, 6, and 12.

The proper divisor sum excludes 12 itself:

The distinction matters. The full divisor sum is used in number-theory formulas, while the proper divisor sum determines whether a number is perfect, abundant, or deficient.

What a Divisor List Reveals

A divisor list contains more information than a set of factors. Its size, pairing pattern, and proper-divisor sum expose several properties of the number.

Prime powers

A prime power has the form p^a. Its divisors follow a simple chain:

For example, 81 = 3⁴, so its positive divisors are 1, 3, 9, 27, and 81. A prime power always has a + 1 positive divisors.

Semiprimes

A semiprime is the product of two primes, allowing the two primes to be equal. If n = pq with different primes p and q, its divisors are 1, p, q, and pq. For example, 77 = 7 × 11 has four positive divisors.

If the primes are equal, the number is a square such as 49 = 7², with divisors 1, 7, and 49.

Special Inputs

The number 1

The number 1 has one positive divisor: itself. It is neither prime nor composite. Its proper-divisor list is empty, so its proper divisor sum is 0.

The number 0

Zero does not have a finite divisor list. Every nonzero integer divides 0 because 0 divided by any nonzero integer is 0.

Negative integers

For a negative input, the positive divisor list is based on its absolute value. The positive divisors of −18 are 1, 2, 3, 6, 9, and 18. Its full integer divisor set also contains their negative counterparts. The displayed divisor count and sums use these positive divisors of the absolute value.

The prime factorization keeps the sign visible:

Prime inputs

A prime input produces one factor pair, 1 × p, and a divisor count of 2. No additional positive divisor exists between 1 and the prime.

Divisors, Factors, and Multiples

Divisor and factor usually name the same values when discussing integers. If 3 divides 18, then 3 is both a divisor and a factor of 18.

A multiple points in the other direction. Since 18 = 3 × 6, the number 18 is a multiple of 3 and a multiple of 6.

A nonzero integer has finitely many divisors but infinitely many multiples. This is one of the simplest ways to keep the terms separate.

Worked Examples

Example: finding the divisors of 84

The factor pairs up to √84 are 1 × 84, 2 × 42, 3 × 28, 4 × 21, 6 × 14, and 7 × 12. Combining both sides of the pairs and sorting them gives:

1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

The prime factorization is 84 = 2² × 3 × 7. Its divisor count is therefore (2 + 1)(1 + 1)(1 + 1) = 12.

How the Calculator Finds Divisors

The calculator validates the input as an integer, separates the sign, and handles 0 and 1 as special cases. For other values, it tests possible divisors from 1 through the square root of the absolute value.

When a tested value divides the number exactly, the calculator records both members of the factor pair. It adds the square root only once when the input is a perfect square, sorts the results, and then calculates the count and sums.

This method is fast for the range supported by the on-page tool. It also avoids the common mistake of scanning every integer up to the input value.

Common Divisor Errors

Leaving out 1 or the number itself

Both belong to the positive divisor list for every positive integer. A proper-divisor list is different because it excludes the number itself.

Listing only the smaller half of each pair

Finding 3 as a divisor of 24 also identifies 8, since 3 × 8 = 24. Testing only up to the square root works only when the paired values are also recorded.

Counting a square root twice

For 49, the pair 7 × 7 contributes one distinct divisor, not two. The positive divisors are 1, 7, and 49.

Calling a decimal quotient a divisor

A divisor must produce an integer quotient. Since 20 ÷ 8 = 2.5, the number 8 is not a divisor of 20.

Treating 1 as prime

A prime must have exactly two positive divisors. The number 1 has only one, so it does not meet the definition.