The factoring calculator has three main parts: a number input box at the top, two radio buttons for selecting the factoring type, and Calculate/Reset buttons at the bottom. Start by clicking in the input box labeled "Enter a number to find its factors" and type any whole number you want to factor.
The calculator accepts both positive and negative integers. Try entering simple numbers like 12, 24, or 50 to see how factoring works. You can also enter larger numbers like 144, 1000, or even 999999—the calculator handles them all instantly.
After entering your number, select your factoring type using the radio buttons. Prime Factoring breaks your number down into prime number building blocks. Complete Factoring finds every number that divides evenly into your input. Each mode gives different information, so choose based on what you need to learn.
Click the blue Calculate Factors button to see your results appear below. The calculator displays factors as a list, and for complete factoring, it also shows factor pairs in multiplication format. Use the Reset button anytime to clear everything and start fresh with a new number.
Using Prime Factoring Mode
Select the Prime Factoring radio button to break your number into its prime building blocks. Prime factoring shows only the prime numbers that multiply together to create your input. For example, enter 12 and click Calculate to see the prime factors: 2,2,3.
This means 12=2×2×3. The calculator lists each prime factor separately, even if it appears multiple times. Try 18 to get 2,3,3 (because 18=2×3×3), or 30 to get 2,3,5 (because 30=2×3×5).
Prime factoring works best when you need to find the greatest common factor (GCF) or least common multiple (LCM) of numbers, simplify radicals, or understand a number's fundamental structure. It's essential for algebra, number theory, and cryptography.
For prime numbers themselves (like 7, 13, or 29), the calculator returns only the number itself as the prime factor. That's because prime numbers cannot be broken down further—they're already in their simplest form. Try entering 17 to see this in action.
Using Complete Factoring Mode
Switch to the Complete Factoring radio button to find all numbers that divide evenly into your input. Complete factoring lists every factor from smallest to largest, including 1 and the number itself. Enter 12 and calculate to see all factors: 1,2,3,4,6,12.
The complete factors list shows you every possible divisor. These are all the whole numbers that go into 12 with no remainder. Try 24 to get eight factors: 1,2,3,4,6,8,12,24. Or enter 36 to find nine factors: 1,2,3,4,6,9,12,18,36.
Complete factoring helps when working with divisibility, fractions (finding common denominators), arrays (arranging items in rows and columns), or area problems (finding rectangle dimensions). It shows all possible ways to divide or group items.
Perfect squares like 16, 25, or 36 have an odd number of factors. This happens because one factor (the square root) pairs with itself. Enter 16 to see factors 1,2,4,8,16—notice how 4×4=16 creates the middle factor.
Understanding Factor Pairs
When you use Complete Factoring mode, the calculator displays factor pairs below the factors list. Factor pairs show two numbers that multiply together to equal your input. For 12, you'll see: 1×12, 2×6, and 3×4.
Each pair represents a different way to arrange 12 items into a rectangle. The pair 3×4 means 3 rows of 4 items, or 4 rows of 3 items. Try 20 to see factor pairs: 1×20, 2×10, and 4×5—all different rectangular arrangements.
Factor pairs are incredibly useful for area calculations. If a rectangle has area 24 square meters, its dimensions could be 1×24, 2×12, 3×8, or 4×6. Enter 24 to see all four possibilities instantly.
The pairs always multiply to give your original number. Verify this by checking: does 3×4 really equal 12? Does 2×6 equal 12? Yes! This symmetry makes factor pairs perfect for checking your work and understanding multiplication relationships.
Working with Negative Numbers
The calculator handles negative numbers by showing both positive and negative factors. Enter −12 in complete factoring mode to see factors: −12,−6,−4,−3,−2,−1,1,2,3,4,6,12. Notice you get twice as many factors—positive and negative versions of each.
Negative factors work because multiplying two negatives gives a positive: (−3)×(−4)=12. But you can also have one negative and one positive: (−6)×2=−12 or 3×(−4)=−12. The calculator lists all factors that divide evenly into your number.
For prime factoring with negative numbers, you'll see an error message: "Cannot find prime factors of negative numbers." This is a mathematical convention—prime factorization is defined only for positive integers. If you need prime factors of a negative number, remove the negative sign first.
Try −20 in complete factoring to see: −20,−10,−5,−4,−2,−1,1,2,4,5,10,20. The factor pairs would include combinations like (−4)×5 and (−2)×10, showing multiple ways to multiply to −20.
Reading Results and Error Messages
After clicking Calculate, your results appear in the results container below the buttons. For prime factoring, you'll see "The prime factors of [number] are:" followed by a list. For complete factoring, you'll see "The factors of [number] are:" followed by the list, then "The factor pairs are:" with multiplication expressions.
The calculator validates your input before computing. If you leave the box empty or enter non-numeric characters like letters or symbols, you'll see the error "Please enter a valid number" in red text. Simply enter a whole number to clear the error.
For negative numbers in prime factoring mode, the error "Cannot find prime factors of negative numbers" appears. This isn't a bug—it's mathematically correct. Switch to complete factoring mode or use the positive version of your number instead.
Very large numbers may take a moment to calculate, especially for complete factoring. The calculator works efficiently but needs time to find all factors of numbers like 10000 or larger. Be patient—the results will appear. After reviewing your results, click Reset to clear the display and try a different number.
What Are Factors
Factors are whole numbers that divide evenly into another number with no remainder. If a×b=c, then a and b are both factors of c. For example, since 3×4=12, both 3 and 4 are factors of 12.
Think of factors as building blocks. The number 12 can be built by multiplying 1×12, 2×6, or 3×4. Each of these numbers—1,2,3,4,6,12—is a factor of 12. Every number has at least two factors: 1 and itself.
Factors are different from multiples. Multiples are what you get when you multiply a number: multiples of 3 are 3,6,9,12,15.... Factors go the other direction: factors of 12 are the numbers you multiply to get 12.
For more detailed explanations of factor properties, prime numbers, and divisibility rules, see number theory fundamentals and divisibility concepts.
Prime Factoring vs Complete Factoring
Prime factoring breaks a number into prime number building blocks. Prime factors are the smallest possible factors (except 1). The number 12 has prime factors 2,2,3 because 2×2×3=12, and you cannot break it down further.
Complete factoring finds all numbers that divide evenly into your input, not just primes. For 12, complete factors are 1,2,3,4,6,12. This includes composite numbers (like 4 and 6) that can themselves be factored further.
Use prime factoring when simplifying radicals (12=4×3=23), finding GCF or LCM, or working with fractions. Use complete factoring when finding divisors, calculating area dimensions, or solving divisibility problems.
Prime factoring is unique—every number has exactly one prime factorization (ignoring order). Complete factoring gives you all possibilities. Both are valuable, just for different purposes. See prime number theory and greatest common factor concepts for deeper understanding.
What Are Prime Numbers
Prime numbers are whole numbers greater than 1 that have exactly two factors: 1 and themselves. The number 7 is prime because only 1 and 7 divide evenly into it. You cannot create 7 by multiplying two smaller whole numbers (other than 1×7).
The first few prime numbers are: 2,3,5,7,11,13,17,19,23,29.... Notice that 2 is the only even prime—all other even numbers are divisible by 2, so they have at least three factors (1, 2, and themselves).
Numbers that aren't prime are called composite numbers. The number 12 is composite because it has many factors: 1,2,3,4,6,12. Composite numbers can be broken down into prime factors: 12=2×2×3.
The number 1 is special—it's neither prime nor composite. By definition, primes must have exactly two factors, but 1 has only one factor (itself). For comprehensive coverage of prime numbers, see prime number theory and prime number lists.