numbers

Factor Calculator | List All Factors And Prime Factorization

Quick reference

Factors of common numbers:

Number	Factors	Prime factorization
12	1,2,3,4,6,12	2^2 x 3
24	1,2,3,4,6,8,12,24	2^3 x 3
36	1,2,3,4,6,9,12,18,36	2^2 x 3^2
100	1,2,4,5,10,20,25,50,100	2^2 x 5^2

functions Shows the working, not just the answer

For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

calculate Accurate to the spec

Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.

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Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.

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Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

Interpretation guide

Factor terms:

Term	Meaning
Factor	A number that divides evenly (no remainder)
Factor pair	Two factors that multiply to the number
Prime factor	A factor that is itself prime
Prime factorization	The number written as a product of primes

lightbulb Worked example — factoring 72

A student needs every factor of 72 and its prime factorization to simplify the fraction 48/72.

Result: Test divisors up to √72 ≈ 8.49: 72 is divisible by 1, 2, 3, 4, 6 and 8, giving the pairs (1,72), (2,36), (3,24), (4,18), (6,12) and (8,9) — twelve factors in total. Prime factorization: 72 = 8 × 9 = 2³ × 3².

What this means: The prime factors reveal the greatest common divisor with 48 (= 2⁴ × 3): the shared primes are 2³ × 3 = 24, so 48/72 simplifies to 2/3. Listing factors up to the square root is enough because every divisor above √72 pairs with one below it — that halving is what makes the method fast.

Formula & methodology

Formula: f is a factor of n if n / f leaves no remainder; prime factorization breaks n into prime building blocks

How factors are found

A factor divides the number with no remainder. The method only needs to test divisors up to the square root, because every divisor above √n pairs with one below it.

Test each integer from 2 up to √n; each hit yields a factor pair (d, n÷d).
Collect the pairs into the full factor list, including 1 and n itself.
For prime factorization, repeatedly divide out the smallest prime factor (the factor-tree method) until only primes remain.

Example: 36 → pairs (1,36), (2,18), (3,12), (4,9), (6,6); prime factorization 36 = 2² × 3². Primes: a prime number’s only factors are 1 and itself, and its factorization is just the number.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How do I find all the factors of a number?

Enter the required values into the input fields and press the calculate button. The tool applies the formula f is a factor of n if n / f leaves no remainder; prime factorization breaks n into prime building blocks and returns the result with a short explanation of each step, so you can check the working yourself.

What is prime factorization?

The Factor Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Lists all the factors of any whole number and its prime factorization. Enter a number and the tool returns every factor pair, the complete list of factors, and the prime factorization using the factor-tree method, with the working shown. Ideal for There is no limit on usage, and it works on any device.

What are factor pairs?

A factor pair is two numbers that multiply to the original value — for 36, the pairs are 1×36, 2×18, 3×12, 4×9, and 6×6. Every positive integer has these complementary pairings, and the calculator lists them all. Perfect squares have a special pair where both factors are equal, like 6×6 for 36, which is why they have an odd number of factors.

Is 1 a factor of every number?

Yes — 1 divides every integer with no remainder, so it is always a factor, and every positive number is also a factor of itself. That is why the smallest possible factor list has two entries. In prime factorization, 1 is special: it is neither prime nor composite and never appears in the factorization, since multiplying by 1 does not change the value.

How is factoring used in fractions?

Factoring is the engine of fraction simplification: dividing the numerator and denominator by their shared factors — the greatest common factor — reduces the fraction to lowest terms, like 12/18 → 2/3. It also finds common denominators for addition and subtraction. Prime factorization makes both operations systematic: the shared primes at their lowest powers are the GCF.

How do I use prime factorization to simplify fractions?

Factor both numbers, then cancel the shared primes. 48 = 2⁴ × 3 and 72 = 2³ × 3² share 2³ × 3 = 24, so 48/72 = (2 × 24)/(3 × 24) = 2/3. The shared prime product is the greatest common divisor — the largest number that divides both.

What is the difference between factors and multiples?

Factors divide INTO the number (the factors of 12 are 1, 2, 3, 4, 6, 12 — a finite list), while multiples are what you get multiplying BY it (12, 24, 36, … — infinite). Mixing them up is common: simplifying fractions needs factors, while finding common denominators needs multiples (specifically the least common multiple).

How can factors help find the least common denominator?

Factor each denominator into primes, then take the highest power of each prime present. For 8 (= 2³) and 12 (= 2² × 3), the LCD is 2³ × 3 = 24. Prime factorization turns denominator-hunting into a mechanical prime-collection exercise.

Why does the method stop checking at the square root?

Because factors come in pairs that multiply to n, and the two members of each pair sit on opposite sides of √n. Once you pass √n you are just rediscovering pairs you already found — for 72 (√72 ≈ 8.49), finding divisor 8 gives the pair (8,9), and there is nothing new above 9 except 12, 18, 24, 36, 72 themselves.

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