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Factoring Calculator



Factor a number
View
60=22⋅3⋅560 = 2^{2} \cdot 3 \cdot 5
Result
Factorization of 60
60=22⋅3⋅560 = 2^{2} \cdot 3 \cdot 5
Prime factorization
22⋅3⋅52^{2} \cdot 3 \cdot 5
Exponent form
2^2 * 3 * 5
Divisor count τ(n)
12
Sum of divisors σ(n)
168
Type
Abundant (σ-n > n)
Neighbors (click to set)
57
4 div
58
4 div
59
prime
60
12 div
61
prime
62
4 div
63
6 div
Apply a factoring pattern
72−327^2 - 3^2
Step-by-step
1
Start: 72−32\text{Start: } 7^2 - 3^2
2
=(7−3)(7+3)= (7 - 3)(7 + 3)
3
=4⋅10=40= 4 \cdot 10 = 40
GCD & LCM
Number A
60=22⋅3⋅560 = 2^{2} \cdot 3 \cdot 5
&
Number B
84=22⋅3⋅784 = 2^{2} \cdot 3 \cdot 7
GCD
gcd⁡=22⋅3=12\gcd = 2^{2} \cdot 3 = 12
LCM
lcm=22⋅3⋅5⋅7=420\mathrm{lcm} = 2^{2} \cdot 3 \cdot 5 \cdot 7 = 420
a \u00B7 b = gcd \u00B7 lcm
60⋅84=12⋅420=504060 \cdot 84 = 12 \cdot 420 = 5040
How this works

What just happened

Every integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic). For 60:
60=22⋅3⋅560 = 2^{2} \cdot 3 \cdot 5

Divisor count

If n=p1a1⋅p2a2⋯n = p_1^{a_1} \cdot p_2^{a_2} \cdots, the number of divisors is the product (a1+1)(a2+1)⋯(a_1+1)(a_2+1)\cdots. Each exponent contributes one more choice.
τ(60)=3⋅2⋅2=12\tau(60) = 3 \cdot 2 \cdot 2 = 12

Worth knowing

Abundant
Proper divisors sum to more than 60 (108). The smallest abundant number is 12.
Nearby numbers
nPrime factorizationdiv #
555⋅115 \cdot 114
5623⋅72^{3} \cdot 78
573⋅193 \cdot 194
582⋅292 \cdot 294
595959prime2
6022⋅3⋅52^{2} \cdot 3 \cdot 512
616161prime2
622⋅312 \cdot 314
6332⋅73^{2} \cdot 76
64262^{6}7
655⋅135 \cdot 134
662⋅3⋅112 \cdot 3 \cdot 118



Getting Started with the Factoring Calculator
Using Prime Factoring Mode
Using Complete Factoring Mode
Understanding Factor Pairs
Working with Negative Numbers
Reading Results and Error Messages
What Are Factors
Prime Factoring vs Complete Factoring
What Are Prime Numbers




























Getting Started with the Factoring Calculator

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 1212, 2424, or 5050 to see how factoring works. You can also enter larger numbers like 144144, 10001000, or even 999999999999—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 1212 and click Calculate to see the prime factors: 2,2,32, 2, 3.

This means 12=2×2×312 = 2 \times 2 \times 3. The calculator lists each prime factor separately, even if it appears multiple times. Try 1818 to get 2,3,32, 3, 3 (because 18=2×3×318 = 2 \times 3 \times 3), or 3030 to get 2,3,52, 3, 5 (because 30=2×3×530 = 2 \times 3 \times 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 77, 1313, or 2929), 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 1717 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 11 and the number itself. Enter 1212 and calculate to see all factors: 1,2,3,4,6,121, 2, 3, 4, 6, 12.

The complete factors list shows you every possible divisor. These are all the whole numbers that go into 1212 with no remainder. Try 2424 to get eight factors: 1,2,3,4,6,8,12,241, 2, 3, 4, 6, 8, 12, 24. Or enter 3636 to find nine factors: 1,2,3,4,6,9,12,18,361, 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 1616, 2525, or 3636 have an odd number of factors. This happens because one factor (the square root) pairs with itself. Enter 1616 to see factors 1,2,4,8,161, 2, 4, 8, 16—notice how 4×4=164 \times 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 1212, you'll see: 1×121 \times 12, 2×62 \times 6, and 3×43 \times 4.

Each pair represents a different way to arrange 1212 items into a rectangle. The pair 3×43 \times 4 means 33 rows of 44 items, or 44 rows of 33 items. Try 2020 to see factor pairs: 1×201 \times 20, 2×102 \times 10, and 4×54 \times 5—all different rectangular arrangements.

Factor pairs are incredibly useful for area calculations. If a rectangle has area 2424 square meters, its dimensions could be 1×241 \times 24, 2×122 \times 12, 3×83 \times 8, or 4×64 \times 6. Enter 2424 to see all four possibilities instantly.

The pairs always multiply to give your original number. Verify this by checking: does 3×43 \times 4 really equal 1212? Does 2×62 \times 6 equal 1212? 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 −12-12 in complete factoring mode to see factors: −12,−6,−4,−3,−2,−1,1,2,3,4,6,12-12, -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(-3) \times (-4) = 12. But you can also have one negative and one positive: (−6)×2=−12(-6) \times 2 = -12 or 3×(−4)=−123 \times (-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 −20-20 in complete factoring to see: −20,−10,−5,−4,−2,−1,1,2,4,5,10,20-20, -10, -5, -4, -2, -1, 1, 2, 4, 5, 10, 20. The factor pairs would include combinations like (−4)×5(-4) \times 5 and (−2)×10(-2) \times 10, showing multiple ways to multiply to −20-20.

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 1000010000 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=ca \times b = c, then aa and bb are both factors of cc. For example, since 3×4=123 \times 4 = 12, both 33 and 44 are factors of 1212.

Think of factors as building blocks. The number 1212 can be built by multiplying 1×121 \times 12, 2×62 \times 6, or 3×43 \times 4. Each of these numbers—1,2,3,4,6,121, 2, 3, 4, 6, 12—is a factor of 1212. Every number has at least two factors: 11 and itself.

Factors are different from multiples. Multiples are what you get when you multiply a number: multiples of 33 are 3,6,9,12,15...3, 6, 9, 12, 15.... Factors go the other direction: factors of 1212 are the numbers you multiply to get 1212.

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 11). The number 1212 has prime factors 2,2,32, 2, 3 because 2×2×3=122 \times 2 \times 3 = 12, and you cannot break it down further.

Complete factoring finds all numbers that divide evenly into your input, not just primes. For 1212, complete factors are 1,2,3,4,6,121, 2, 3, 4, 6, 12. This includes composite numbers (like 44 and 66) that can themselves be factored further.

Use prime factoring when simplifying radicals (12=4×3=23\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}), 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 11 that have exactly two factors: 11 and themselves. The number 77 is prime because only 11 and 77 divide evenly into it. You cannot create 77 by multiplying two smaller whole numbers (other than 1×71 \times 7).

The first few prime numbers are: 2,3,5,7,11,13,17,19,23,29...2, 3, 5, 7, 11, 13, 17, 19, 23, 29.... Notice that 22 is the only even prime—all other even numbers are divisible by 22, so they have at least three factors (11, 22, and themselves).

Numbers that aren't prime are called composite numbers. The number 1212 is composite because it has many factors: 1,2,3,4,6,121, 2, 3, 4, 6, 12. Composite numbers can be broken down into prime factors: 12=2×2×312 = 2 \times 2 \times 3.

The number 11 is special—it's neither prime nor composite. By definition, primes must have exactly two factors, but 11 has only one factor (itself). For comprehensive coverage of prime numbers, see prime number theory and prime number lists.