Find every factor of any number in seconds — enter a number and get its complete factor list, factor pairs, divisor count, and number classification. Free, fast, and mobile-friendly.
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A factor of a number is any whole number that divides it with nothing left over. The factors of 12 are 1, 2, 3, 4, 6, and 12, because each of those divides 12 evenly. Listing every factor by hand is slow and it is easy to miss one, especially with larger numbers. Our free Factor Calculator finds the complete factor list of any number up to one hundred million instantly, and it also shows the factor pairs, the total divisor count, and whether the number is prime, composite, perfect, abundant, or deficient.
Factors are the building blocks of most of number theory. Simplifying fractions depends on finding common factors. The greatest common divisor of two numbers is simply the largest factor they share. Scheduling problems, tiling problems, cryptography, and even musical rhythm all reduce at some point to asking which numbers divide which. Having a tool that lists factors reliably means you can focus on the actual problem instead of grinding through trial division on paper.
This calculator tests every possible divisor up to the square root of your number, which is the fastest correct method for complete factor lists. It handles the edge cases gracefully: 1 is reported as neither prime nor composite, prime numbers are identified by having exactly two factors, and perfect numbers like 6 and 28 are flagged when their proper factors add up exactly to the number itself. Whether you are checking homework, simplifying a fraction, or exploring number patterns, you get a complete, trustworthy answer in a fraction of a second.
A factor of a whole number n is a whole number d such that n ÷ d leaves no remainder. Equivalently, d is a factor of n when there exists another whole number k with d × k = n. Every number has at least the trivial factors 1 and itself; numbers with exactly those two factors are called prime, and numbers with additional factors are called composite. The number 1 is the lone exception — it has only one factor, so it is classified as neither prime nor composite.
Factors naturally arrive in pairs. If d divides n, then the quotient n ÷ d is also a factor, and the two multiply back to n. For 36, the pairs are 1 × 36, 2 × 18, 3 × 12, 4 × 9, and 6 × 6. Notice that once the two members of a pair meet (6 × 6), you have found everything — that meeting point is the square root. This is why the calculator only needs to test divisors up to √n: every factor larger than the square root is already paired with a smaller one that was found.
Proper factors are all the factors except the number itself. They reveal a beautiful classification discovered by the ancient Greeks. If the proper factors add up to exactly the number, it is perfect (6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14). If they add to more, the number is abundant (12, since 1 + 2 + 3 + 4 + 6 = 16). If they add to less, it is deficient — and most numbers, including all primes, are deficient. The calculator reports all three sums so you can see this structure directly.
One more useful count: the number of factors tells you how divisible a number is. Highly composite numbers like 36, 48, and 60 have unusually many factors, which is exactly why 60 became the basis of our time and angle systems — 60 splits evenly into halves, thirds, quarters, fifths, sixths, tenths, twelfths, and more. When you need a quantity that plays nicely with many group sizes, you want a number with many factors.
Test each integer d from 1 up to √n: if n ÷ d has remainder 0, then both d and n ÷ d are factors.
You never need to test past the square root — any factor larger than √n pairs with one smaller than √n that you already found.
Compute √36 = 6, so test d = 1 through 6. 36 ÷ 1 = 36 (factors 1, 36). 36 ÷ 2 = 18 (factors 2, 18). 36 ÷ 3 = 12 (factors 3, 12). 36 ÷ 4 = 9 (factors 4, 9). 5 does not divide 36. 36 ÷ 6 = 6 (factor 6, paired with itself). Sorted, the complete list is 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine factors forming five pairs. The proper factors sum to 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55, which exceeds 36, so 36 is abundant.
√100 = 10. Testing 1 through 10: 1 × 100, 2 × 50, 4 × 25, 5 × 20, 10 × 10. Note that 3, 6, 7, 8, and 9 do not divide 100. The full list is 1, 2, 4, 5, 10, 20, 25, 50, 100 — nine factors. Proper-factor sum: 1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 = 117, which is greater than 100, so 100 is abundant as well.
√97 ≈ 9.8, so test 1 through 9. None of 2, 3, 4, 5, 6, 7, 8, 9 divides 97. The only factors are 1 and 97, which proves 97 is prime. This trial-division test is exactly how primality is checked for small numbers — a number with no divisors up to its square root must be prime.
The number 1 has exactly one factor: itself. It is neither prime (which requires exactly two factors) nor composite (which requires more than two). Its proper-factor sum is 0, less than 1, so by the abundance classification it counts as deficient. The calculator reports all of this explicitly instead of forcing 1 into a category where it does not belong.
A community garden is laying out 36 identical square planting beds and wants them in a neat rectangle with no gaps. Which rectangle dimensions are possible? The side lengths must be a factor pair of 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, or 6 × 6. The calculator lists all five options instantly, and the gardeners pick 6 × 6 for the most square-like bed — or 4 × 9 if the plot is long and narrow.
A teacher has 100 identical stickers to divide into equal prize bags with nothing left over. The possible bag counts are the factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, or 100 bags. Ten bags of ten stickers each is the natural choice, but the factor list shows every fair division at a glance — no leftover stickers, no arguments.
Event planners use the same idea in reverse. If 48 guests must sit at identical round tables with every seat filled, the table count has to divide 48: the calculator gives 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Six tables of eight is usually the sweet spot for conversation, and the factor list confirms it divides evenly before a single chair is moved.
Factor pairs are more than a tidy way to display results — they are the reason many classic problems have clean answers. Splitting into equal groups, arranging rectangles, dividing inheritances, and scheduling repeating events all come down to choosing a factor pair. Listing pairs instead of a flat list makes the structure visible: each pair is one complete way to break the number into two whole-number pieces.
There is also a lovely symmetry: reading the pairs of 36 from the outside in (1×36, 2×18, 3×12, 4×9, 6×6), the first elements rise while the second elements fall, and they meet at the square root. For non-squares like 30 (1×30, 2×15, 3×10, 5×6), the pairs never quite meet, and the count stays even. This picture explains at a glance why squares have an odd number of factors and everything else has an even number.
Programmers meet the same pairing idea when optimizing loops: to find all divisors you iterate only to √n and record both d and n÷d, cutting the work from n steps to √n steps. For n = 100,000,000 that is the difference between a hundred million iterations and ten thousand — the reason this calculator answers instantly even at its maximum input.
√1000 ≈ 31.6. Testing 1 through 31 finds divisors 1, 2, 4, 5, 8, 10, 20, 25, and their partners 1000, 500, 250, 200, 125, 100, 50, 40. Sorted: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000 — sixteen factors. Because 1000 = 2³ × 5³, the count also follows the divisor-count rule (3+1)(3+1) = 16.
Divisors up to √28 ≈ 5.3: 1, 2, 4 and partners 28, 14, 7. Full list: 1, 2, 4, 7, 14, 28. Proper factors sum to 1 + 2 + 4 + 7 + 14 = 28 — exactly the number itself, so 28 is perfect, only the second one known after 6. Perfect numbers are rare and were considered almost mystical by the Pythagoreans.
The number 144 has factors 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 — fifteen factors, an odd count, which immediately tells you 144 is a perfect square (12²) even before you compute the root. This odd-count test works for every perfect square and no other numbers.
The divisor-count rule deserves one more look, because it lets you predict how many factors a number has without listing them. Write the number as a product of prime powers, for example 36 = 2² × 3², then add one to each exponent and multiply: (2+1)(2+1) = 9. That matches the nine factors found above. It works because choosing a divisor means independently choosing how many copies of each prime to include — 0, 1, or 2 copies of the 2s (three choices) times 0, 1, or 2 copies of the 3s (three choices). Nine combinations, nine divisors.
List all factors of 60 and state whether it is abundant, deficient, or perfect. (Hint: √60 ≈ 7.7, so test 1 through 7.)
A number has exactly three factors. What must be true about it? (Hint: think about which numbers have an odd factor count, and which of those have exactly three.)
The proper factors of a mystery number include 1, 2, 4, 5, 10, and 20, and they sum to 42. The number itself is 42. What classification does this give it, and what are its remaining factors?
An organizer must split 96 players into equal-sized groups for the opening round, with every group the same size and nobody left out. The allowed group sizes are exactly the factors of 96. Running the calculator gives: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 — twelve options forming six pairs (1×96, 2×48, 3×32, 4×24, 6×16, 8×12).
Practical constraints narrow it down fast. Groups of 1 or 96 are meaningless, and groups above 16 make scheduling unwieldy, leaving 2, 3, 4, 6, 8, 12, and 16. The organizer wants groups large enough for round-robin play but small enough to finish in an evening, so 8 groups of 12 wins: twelve players per group is big enough for variety, and eight groups fit the eight available courts. The factor list turned a vague debate into a two-minute decision backed by every mathematically possible option.
The Factor Calculator lists every factor of a whole number (up to 100,000,000) by testing divisors up to the square root, then reports the factor pairs, divisor count, factor sum, proper-factor sum, and classifications: prime vs composite from the factor count, and perfect vs abundant vs deficient from the proper-factor sum.
The factors of a number are all the whole numbers that divide it with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because 12 ÷ each of these leaves no remainder. Every factor pairs with another (1×12, 2×6, 3×4).
Test every integer from 1 up to the square root of the number. Whenever one divides evenly, record both it and the quotient as factors. For 36, √36 = 6, so testing 1 through 6 finds every factor: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Factors divide into a number and are less than or equal to it (factors of 12: 1, 2, 3, 4, 6, 12). Multiples are produced by multiplying the number and are greater than or equal to it (multiples of 12: 12, 24, 36, 48, and so on). Factors are finite; multiples go on forever.
A prime number has exactly two factors: 1 and itself. If your factor list contains anything else, the number is composite. The number 1 is the exception — with only one factor, it is neither prime nor composite.
Add up the proper factors (all factors except the number itself). If the sum equals the number, it is perfect (6, 28). If the sum is greater, it is abundant (12). If the sum is smaller, it is deficient — which includes all prime numbers and most composites.
Factors pair up as d × (n÷d), but for a perfect square the square root pairs with itself (6×6 = 36), contributing one unpaired factor. So squares always have an odd factor count, and only squares do — an odd count is a perfect-square test.
Factor pairs list every way to split a number into two whole-number pieces. They answer questions like what rectangle sizes use exactly 36 tiles (1×36 through 6×6) or how 48 guests can be seated at equal tables — any pair gives a valid arrangement.
Trial division up to the square root is the fastest simple method: for n = 100,000,000 you test only 10,000 candidates instead of 100,000,000. Recording both d and n÷d each time a divisor is found captures the complete list. This calculator uses exactly that method.
Cite this page as: Factor Calculator. Sigma Calculator. Retrieved October 9, 2026, from https://sigmacalculator.net/math-calculators/factor-calculator/
Last updated: October 9, 2026