Divisor - Factors, Common Divisors, and GCD

A divisor is a whole number that divides another whole number with no remainder, and this tool lists every positive factor of any integer you type in. Enter a single number to get the full sorted list, the count, and the sum, and add a second number for shared factors and the GCD.

• • Homework and self-check: Type a number to see every positive factor at once, which is faster than running the divisibility rule by hand.
• • Shared-factor and GCD work: Add a second positive integer to get the shared factor list and the GCD without retyping the factor list for each number.
• • Factor pairs and prime checks: Use the factor-pair table to see both halves of every pair, and confirm a prime by reading a count of 2.
• • Practical splitting and grouping: Read the factor list to see whether a total splits evenly into the group size you have in mind, and which group sizes give an exact split.

The word 'divisor' is interchangeable with 'factor' in number theory. The integer you are factorizing is sometimes called the dividend, and the result of dividing by a factor is the quotient.

If you only need to know whether one number is divisible by another, our divisibility test calculator applies the n mod d = 0 check directly and lists the matching one-line rule.

The calculator uses a square-root limit: it tests every integer d from 1 up to floor(sqrt(n)) and, whenever d divides n, records d and the partner n / d as a factor pair. This cuts the work in half and still returns the full list once sorted.

• n: The first positive integer whose factors are being listed. Defaults to 60, which has twelve positive factors and shows the full result panel on the first try.
• m: The optional second positive integer. Leave at 0 to list factors of n only; enter any positive integer to enable shared factors and the GCD.
• d: A candidate factor. The loop tests d = 1, 2, 3, ... up to floor(sqrt(n)) and pairs each successful d with n / d.
• GCD: Greatest common factor of n and m. Reported only when m is greater than 0 and computed as the largest shared factor of the two numbers.

The same square-root loop runs on both numbers when a second integer is supplied. The GCD is the largest element of the intersection, which the calculator reports alongside the count and the full list. As Wolfram MathWorld states, a factor of an integer n is any integer d for which n = d * k, and the calculator uses that exact definition to label every entry.

According to Wolfram MathWorld, a divisor of an integer n is any integer d for which n = d * k for some integer k, and the calculator uses that exact definition when deciding which numbers to include in the list.

When the calculator reports that n mod d equals 0 for a specific d, you can drop the same pair into our modulo calculator to read the remainder for any positive divisor without re-typing the number.

Four small ideas cover everything the calculator can show.

These four ideas cover every output the calculator shows. The square-root limit tests only half of the search space, which is why the page is faster than dividing by every integer up to n.

When the factor list is long and you want the exponent form behind it, the prime factorization calculator returns the prime factors of the same number in a single step.

Five short steps cover the single-number and the two-number workflows.

1. 1 Enter the first number: Type any positive integer in the 'Number' field. The default 60 has twelve positive factors, so the result panel is populated on the first try.
2. 2 Add a second number (optional): Type a second positive integer in the 'Second Number (optional)' field to enable the shared list and the GCD. Leave the field at 0 to skip that block.
3. 3 Read the list and count: The result panel shows the sorted list, the count, and the sum. The list is in ascending order, with the smallest entry (1) on the left and the largest (n) on the right.
4. 4 Read the shared factors and GCD: When the second number is greater than 0, the panel also shows the shared list, the shared count, and the GCD. The GCD is the largest step that fits both numbers.
5. 5 Scan the factor-pair table: The factor-pair table below the result panel shows both halves of every pair, so you can see how each entry d maps to n / d. For a prime number, the table reduces to the single pair (1, n).

Enter 36 in the first number field and 24 in the second. The result panel reads 'Factor count: 9, sum: 91, list: 1, 2, 3, 4, 6, 9, 12, 18, 36' and the shared-factor block reads 'Shared factors: 1, 2, 3, 4, 6, 12, count: 6, GCD: 12'. Change the second number to 0 to see only the factors of 36.

If you have two numbers and want the largest shared factor only, the greatest common factor calculator returns the GCD in a single field and is the natural follow-up once the shared-factor block is visible here.

A purpose-built tool is faster and more reliable than running the divisibility rule for each small integer by hand.

• • Full sorted list in one step: The calculator returns the complete ascending list of positive factors, the count, and the sum in a single click, which saves running the divisibility rule by hand.
• • Shared factors and GCD without retyping: Adding a second number switches on the shared list and the GCD, so you do not need to write down the factor list for each number before intersecting them.
• • Factor-pair table makes the structure visible: The factor-pair table shows both halves of every pair, which makes it easy to see that a prime reduces to (1, n) and that a perfect square has one pair where d = n / d.
• • Works on large inputs: The square-root limit keeps the loop fast up to 15 digits, and the inputs are clamped to the JavaScript safe-integer range so the result is always exact for everyday classroom and textbook numbers.

When the factor count is long, the prime factorization calculator returns the exponent form in a single step, which is the same logic that powers the textbook rule 'add 1 to each exponent and multiply'.

When you want the same list with the count of factors and a quick prime-factorization hint, the factor calculator presents the same data in a more classroom-ready layout.

The factor cards below describe the structural properties of the number you typed, in the same order the live factor-pair table renders them.

• • The calculator accepts positive integers up to 15 digits (capped at Number.MAX_SAFE_INTEGER). Inputs above that cap are clamped to keep the result exact.
• • Negative integers and zero are not supported. The form rejects n less than 1.

When the factor list is long, the prime factorization calculator gives the exponent form in one step. As Wolfram MathWorld states, the greatest common factor of two positive integers is the largest integer that divides both, which the calculator reports as the GCD.

According to Wolfram MathWorld, the greatest common factor of two positive integers is the largest integer that divides both, and the calculator reports that exact value as the GCD when a second number is supplied.

When the factor count on the result panel is exactly 2, the prime number checker confirms the primality reading and lists the nearest prime neighbors for context.