A factor of a number is a whole number that divides it exactly, leaving no remainder. This Factor Calculator lists every factor of your number, arranges them into factor pairs, counts and sums them, and shows the prime factorization underneath — the complete divisibility picture in one click.
Worked Example: The Factors of 36
For 36 the calculator lists 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine factors — arranged in the pairs (1, 36), (2, 18), (3, 12), (4, 9) and (6, 6), each multiplying back to 36. The factor sum is 91, and the prime factorization underneath reads 36 = 2² × 3². Notice the count is odd: that happens exactly for perfect squares, because the middle pair (6, 6) pairs a number with itself. Notice also how the search only needs to run up to √36 = 6 — every divisor found below the square root brings its partner above it for free, which is why factoring even large numbers by hand stays feasible far longer than students expect.
How Factoring Works
To factor n, test each whole number d from 1 up to √n: if n ÷ d leaves no remainder, both d and n/d are factors. The square-root stopping point is the whole trick — divisors come in pairs straddling √n, so the second half of the list is just the partners of the first half read backwards. The prime factorization is the deeper fingerprint: every whole number greater than 1 is either prime or a unique product of primes (the fundamental theorem of arithmetic). From 36 = 2² × 3² the factor count follows without listing: add 1 to each exponent and multiply, (2+1)(2+1) = 9. The same exponents power the greatest common factor and least common multiple of two numbers — take minima for the GCF, maxima for the LCM.
Where Factors Matter
Simplifying fractions is factoring in disguise: 36/48 reduces by the shared factor 12 to 3/4. Tiling and grouping problems — how many equal rows fit 36 chairs, which rectangle dimensions have whole-number sides — are factor-pair questions word for word. Cryptography runs the search backwards: multiplying two large primes is instant, but recovering them from the product (factoring) is so hard at 300-digit scale that RSA encryption bets the internet on it. And in music and scheduling, common factors of two periods tell you when patterns realign — the same LCM logic as gears and blinking lights.
Common Mistakes
Forgetting 1 and the number itself, which are always factors. Stopping the search too early — testing only to n/2 wastes effort but finds everything; stopping before √n misses pairs. Confusing factors with multiples: factors of 36 are at most 36, multiples of 36 are at least 36, and mixing the directions wrecks fraction work. Finally, assuming an odd factor count signals an error — it signals a perfect square instead.
Counting and Summing Factors Without Listing Them
The prime factorization answers counting questions directly. From 36 = 2² × 3², the factor count is (2+1)(2+1) = 9, and the factor sum has a formula too: multiply, over each prime, the geometric sum of its powers — (1 + 2 + 4)(1 + 3 + 9) = 7 × 13 = 91, matching the sum this calculator prints. The reason is clean: every factor of 36 is 2^a × 3^b with a in {0,1,2} and b in {0,1,2}, and expanding that product generates each factor exactly once. The factor sum unlocks a classical classification: compare the sum of proper factors (all except the number itself) with the number. For 36 the proper sum is 91 − 36 = 55, greater than 36, so 36 is abundant. For 28 the factors are 1, 2, 4, 7, 14, 28, the proper sum is exactly 28, and 28 is perfect — the second of the famous sequence 6, 28, 496, 8128 that Euclid already knew how to generate. Numbers whose proper sum falls short, like any prime (proper sum 1), are deficient. Enter 28 or 496 above and watch the count and sum line confirm the classification in one step.
Factors, GCF and LCM as One Story
Two numbers’ factor lists overlap in their common factors, the greatest of which is the GCF; their multiples overlap in common multiples, the least of which is the LCM. The prime-exponent view unifies the three pages of this little family: list each number’s exponents, take the minimum exponent per prime for the GCF and the maximum for the LCM, and note the elegant identity GCF(a, b) × LCM(a, b) = a × b, which holds because min + max = the sum of the two exponents. With 36 = 2²·3² and 48 = 2⁴·3¹: GCF = 2²·3 = 12, LCM = 2⁴·3² = 144, and 12 × 144 = 1728 = 36 × 48, checking the identity. Factor lists are the brute-force version of the same picture — slower, but the version that builds the intuition the shortcuts stand on.
Related Tools
Continue with the GCF Calculator, the Prime Factorization Calculator, the Absolute Value Calculator, or browse the full Math Calculators hub.
Frequently Asked Questions
What are the factors of 36?
1, 2, 3, 4, 6, 9, 12, 18 and 36 — nine factors, in the pairs (1, 36), (2, 18), (3, 12), (4, 9) and (6, 6).
How many factors does a prime number have?
Exactly two: 1 and itself. That is the definition of prime, and this calculator shows the count so you can check primality at a glance.
Why do perfect squares have an odd number of factors?
Factors pair up as (d, n/d). For a square, the pair at d = √n pairs the number with itself, adding one unpaired factor — so the total is odd.
How can I get the factor count from prime factorization?
Add 1 to every exponent and multiply the results. For 36 = 2² × 3² that is (2+1)(2+1) = 9 factors.
What is the difference between factors and multiples?
Factors divide into the number and are never larger than it; multiples are what you get multiplying the number, and are never smaller (for positive numbers).
Does this work for negative numbers?
Factor lists are conventionally given for the positive value; the factors of −36 are the factors of 36 with signs attached in pairs. Enter the positive number here.