All factors and prime factorization.
d is a factor of n ⟺ n mod d = 0, for all integers d where 1 ≤ d ≤ n
A factor of n is any positive integer d such that n ÷ d results in a whole number (i.e., the remainder is 0). To find all factors efficiently, the calculator checks every integer from 1 up to the square root of n. When a divisor d is found, its pair n ÷ d is also recorded. This means you never need to check beyond √n, making the process fast even for large numbers.
Check all integers d from 1 to √36 = 6: • d=1: 36 ÷ 1 = 36 ✓ → factors: 1 and 36 • d=2: 36 ÷ 2 = 18 ✓ → factors: 2 and 18 • d=3: 36 ÷ 3 = 12 ✓ → factors: 3 and 12 • d=4: 36 ÷ 4 = 9 ✓ → factors: 4 and 9 • d=5: 36 ÷ 5 = 7.2 ✗ → not a factor • d=6: 36 ÷ 6 = 6 ✓ → factor: 6 (pair is itself) Collect and sort all unique factors found.
Result: All factors of 36: **1, 2, 3, 4, 6, 9, 12, 18, 36** (9 factors total)
The calculator returns every positive integer that divides 36 without a remainder. There are 9 factors in total: 1, 2, 3, 4, 6, 9, 12, 18, and 36. They appear in ascending order. Notice that 36 is a perfect square, which is why it has an odd number of factors — the square root (6) pairs with itself. The prime factorization of 36 is 2² × 3², which also lets you calculate the total factor count as (2+1) × (2+1) = 9.
What is a factor? A factor (also called a divisor) of a whole number n is any positive integer that divides n evenly — meaning the division produces no remainder. Every number n has at least two factors: 1 and itself.
Factor pairs Factors naturally come in pairs. For n = 36, the pairs are (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6). Each pair multiplies to equal 36. Pair-based searching is why checking only up to √n is sufficient.
Prime vs. composite numbers
How many factors does a number have? If the prime factorization of n is p₁^a × p₂^b × p₃^c × …, the total number of factors is (a+1)(b+1)(c+1)… Example: 36 = 2² × 3² → (2+1)(2+1) = 9 factors.
Common uses of factoring
The only factor of 1 is 1 itself. It is the only positive integer with exactly one factor, which is why 1 is classified as neither prime nor composite.
Divide the number sequentially by integers starting from 1 up to its square root. Any integer that divides without a remainder is a factor; record both that divisor and its quotient as a factor pair. Combine and sort all results.
Prime factorization expresses a number as a product of prime numbers (e.g., 60 = 2² × 3 × 5). Every factor of the number can be constructed by choosing exponents for each prime between 0 and its maximum in the factorization, making prime factorization the backbone of finding all factors.
In standard elementary and middle-school mathematics, factors are considered positive integers. In higher algebra, integer factors can be negative (e.g., −3 is an integer factor of 12), but for most practical purposes — simplifying fractions, GCF, LCM — only positive factors are used.
Divisibility rules are shortcuts to test whether common numbers (2, 3, 5, etc.) are factors of n. For example, if n ends in 0 or 5, then 5 is a factor; if the sum of its digits is divisible by 3, then 3 is a factor. These rules speed up manual factoring.
A factor divides evenly INTO a number, while a multiple is the result of multiplying a number by a positive integer. For example, factors of 6 are 1, 2, 3, and 6; multiples of 6 are 6, 12, 18, 24, etc.
Every positive integer greater than 1 has at least two factors: 1 and itself. The number 1 is the only positive integer with exactly one factor (just itself). Numbers with exactly two factors are called prime numbers.
Yes. The calculator checks divisors up to √n, so it handles large numbers efficiently. However, extremely large numbers (with hundreds of digits) may take longer due to the computational steps involved.
A number has an odd number of factors if and only if it is a perfect square. This is because most factors pair up, but the square root pairs with itself, adding a single (unpaired) factor. For example, 25 = 5² has three factors: 1, 5, 25.
To find the GCF of two numbers, you can list all factors of each and identify the largest factor they share. For example, factors of 18 are {1,2,3,6,9,18} and factors of 24 are {1,2,3,4,6,8,12,24}; the GCF is 6.
Only sources that have been reviewed are shown. Unverified citations are never published.
Spotted an inaccuracy or have an idea to improve this calculator? Let us know.
We use cookies to operate the site, and — with your consent — to show advertising and measure usage. You can change your choice anytime in Cookie Settings.