The MathBlog factoring calculator helps you quickly find all factors of a given number. Our tool will calculate the factors, prime factors, and factor pairs of a number you input. It also gives a detailed factor tree visualization, making it easy to see the step-by-step breakdown of how the number is factored into its prime components.
Factoring Calculator
What are factors of a number?
Factors of a number are the integers that divide the number exactly without leaving a remainder. In other words, if a number 𝑛 can be divided by another integer 𝑥 such that the result is also an integer ((a number that is not a fraction; a whole number), then 𝑥 is a factor of 𝑛.
What are prime factors?
Prime factors of a number are the prime numbers that multiply together to give the original number. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. To find the prime factors of a number, you break down the number into its prime components.
Prime factorization is useful for:
- Finding the Greatest Common Divisor (GCD) of two numbers.
- Finding the Least Common Multiple (LCM) of two numbers.
- Simplifying fractions by canceling out common prime factors.
- Cryptography and understanding number properties.
Prime factors reveal the “building blocks” of a number, making it easier to work with in many areas of mathematics. For a visual representation, we can use the factor tree of 12 to make it easier to understand.
| 12 | ||
| | | \ | |
| 2 | 6 | |
| | | \ | |
| 2 | 3 |
That tells us that \( 12 = 2 \times 2 \times 3 \).
That means the prime factors of 12 are 2 and 3.
What is a common factor?
A common factor is a factor that two or more numbers share. In other words, it’s a number that divides each of the given numbers exactly, without leaving a remainder. To find the common factors of multiple numbers, you first determine the factors of each number individually and then identify the factors that are common to all.
What are factor pairs?
Factor pairs are pairs of numbers that, when multiplied together, produce a given number. Each factor pair consists of two factors whose product equals the original number. They are useful for understanding the structure of a number and its divisors.
The properties of factor pairs are:
- Symmetry: Factor pairs mirror around the square root of the number. For example, if n=24, pairs like (1, 24) and (24, 1) are symmetrical.
- Product Equals the Number: Each factor pair’s product must always equal the original number.
- Positive and Negative Pairs: Factor pairs can be positive or negative. For example, for 24, both (3, 8) and (-3, -8) are valid factor pairs.
Divisibility rules
Divisibility rules are shortcuts that help determine whether a number can be divided by another number without performing long division. They’re useful to quickly identify factors and simplify math problems. Here are some basic divisibility rules that you should know:
- Divisibility by 2:
- A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
- Example: 84 is divisible by 2 because the last digit (4) is even.
- Divisibility by 3:
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- Example: 123 is divisible by 3 because the sum of its digits (1 + 2 + 3 = 6) is divisible by 3.
- Divisibility by 4:
- A number is divisible by 4 if the last two digits form a number that is divisible by 4.
- Example: 316 is divisible by 4 because the last two digits (16) are divisible by 4.
- Divisibility by 5:
- A number is divisible by 5 if its last digit is 0 or 5.
- Example: 135 is divisible by 5 because it ends in 5.
- Divisibility by 6:
- A number is divisible by 6 if it is divisible by both 2 and 3.
- Example: 84 is divisible by 6 because it is divisible by 2 (ends in 4) and by 3 (sum of digits is 12, which is divisible by 3).
- Divisibility by 8:
- A number is divisible by 8 if the last three digits form a number that is divisible by 8.
- Example: 1,024 is divisible by 8 because the last three digits (024) are divisible by 8.
- Divisibility by 9:
- A number is divisible by 9 if the sum of its digits is divisible by 9.
- Example: 729 is divisible by 9 because the sum of its digits (7 + 2 + 9 = 18) is divisible by 9.
- Divisibility by 10:
- A number is divisible by 10 if its last digit is 0.
- Example: 340 is divisible by 10 because it ends in 0.
Additional Divisibility Rules
- Divisibility by 11:
- A number is divisible by 11 if the difference between the sum of its alternating digits is 0 or a multiple of 11.
- Example: 1,122 is divisible by 11 because (1 + 2) – (1 + 2) = 0.
- Divisibility by 12:
- A number is divisible by 12 if it is divisible by both 3 and 4.
- Example: 144 is divisible by 12 because it is divisible by 3 (sum of digits is 9) and by 4 (last two digits are 44).
How to factor numbers (Factorization)
Factorization (or factoring) is the process of breaking down a number into its prime factors or other factors that multiply together to form the original number. This process is useful for simplifying mathematical expressions, solving equations, and finding common divisors or multiples.
There are three methods to calculate the factors of a number.
1. Basic Factorization
Basic Factorization is a straightforward method of finding all factors of a number by systematically testing each integer from 1 up to the number itself to see which ones divide evenly (with zero remainder).
Here’s how basic factorization works:
- Start with 1 and check divisibility up to the given number n.
- For each number i that divides n without a remainder, record both i and n÷i as factors.
- Continue until all factor pairs are identified.
2. Prime Factorization
Prime Factorization is the process of breaking down a number into its prime factors—the prime numbers that multiply together to produce the original number. A prime number is a number greater than 1 that has no divisors other than 1 and itself. The goal of prime factorization is to express the number as a product of prime numbers.
Here’s how prime factorization works
- Start with the Smallest Prime Number:
- Begin by dividing the number by the smallest prime number (usually 2).
- If it’s divisible, continue dividing by that prime number until it no longer divides evenly.
- Move to the Next Prime Number:
- If the number is no longer divisible by the smallest prime, move to the next prime (3, 5, 7, etc.) and repeat the process.
- Repeat Until the Quotient is 1:
- Continue dividing by prime numbers until the quotient reaches 1.
- List the Prime Factors:
- Collect all the prime numbers you divided by. These are the prime factors of the original number.
3. Factorization Using Trial Division
Factorization using trial division is a systematic method for finding all the factors of a number by testing divisibility with consecutive integers, starting from 1 up to the square root of the number. Unlike other factorization methods, it is not restricted to just prime numbers but considers all integers as potential divisors until the square root.
Here’s how trial division works:
- Find the Square Root: First, determine the square root of the given number \( n \) and round down to the nearest whole number. Let’s call this \( s \).
- Test Consecutive Integers from 1 to \( s \): Start from 1 and test each integer up to \( s \) to see if it divides \( n \) without leaving a remainder.
- Record Factor Pairs: For each integer \( i \) that divides \( n \), both \( i \) and \( n \div i \) are factors of \( n \).
Example Factorization
For the number 12, its factors are: \( 1, 2, 3, 4, 6, \) and \( 12 \). Here’s how you calculate:
Basic Method
- 1 divides 12 exactly, as \( 12 \div 1 = 12 \),
- 2 divides 12 exactly, as \( 12 \div 2 = 6 \),
- 3 divides 12 exactly, as \( 12 \div 3 = 4 \),
- 4 divides 12 exactly, as \( 12 \div 4 = 3 \),
- 6 divides 12 exactly, as \(12 \div 6 = 2 \),
- 12 divides 12 exactly, as \( 12 \div 12 = 1 \).
Prime Factorization
- Start with the smallest prime number, 2: \( 12 \div 2 = 6 \).
- Continue dividing by 2: \( 6 \div 2 = 3 \).
- Next prime number is 3: \( 3 \div 3 = 1 \).
- Stop as 4 would be higher than 3.56.
\(12 = 2 \times 2 \times 3 \) or \( 12 = 2^2 \times 3 \).
The prime factors of 12 are 2 and 3.
Trial Division
- Find the square root of 12, which is \( \sqrt{12} \approx 3.46 \). Round down to 3.
- Test all integers from 1 through 3 for division:
- \( 12 \div 1 = 12 \Rightarrow (1, 12) \),
- \( 12 \div 2 = 6 \Rightarrow (2, 6) \),
- \( 12 \div 3 = 4 \Rightarrow (3, 4) \).
The factor pairs are \( (1, 12), (2, 6), \) and \( (3, 4) \).
The factors of 12 are \( 1, 2, 3, 4, 6, \) and \( 12 \).
Prime Factorization vs Trial Division Method
- Prime Factorization Method: When using prime factorization, you only test divisibility using prime numbers such as 2, 3, 5, 7, 11, and so on. You keep dividing n by prime numbers until n is fully broken down into a product of only prime numbers.
- Trial Division Method: Trial division tests every integer, not just prime numbers, up to s. The goal is to find all factors of n (including composite factors), not just the prime factors. For example, if using trial division for 12, you would test 1, 2, and 3 (all integers up to its square root, \( \sqrt{12} \approx 3.46 \) ) to get factor pairs: (1, 12), (2, 6), and (3, 4).
Factors of 1-100
You can use the MathBlog factor calculator at the top of the to obtain the list of factors for any number, or you could refer to our table below for factors of numbers 1 to 100.
| Factors of 1 | 1 |
| Factors of 2 | 1, 2 |
| Factors of 3 | 1, 3 |
| Factors of 4 | 1, 2, 4 |
| Factors of 5 | 1, 5 |
| Factors of 6 | 1, 2, 3, 6 |
| Factors of 7 | 1, 7 |
| Factors of 8 | 1, 2, 4, 8 |
| Factors of 9 | 1, 3, 9 |
| Factors of 10 | 1, 2, 5, 10 |
| Factors of 11 | 1, 11 |
| Factors of 12 | 1, 2, 3, 4, 6, 12 |
| Factors of 13 | 1, 13 |
| Factors of 14 | 1, 2, 7, 14 |
| Factors of 15 | 1, 3, 5, 15 |
| Factors of 16 | 1, 2, 4, 8, 16 |
| Factors of 17 | 1, 17 |
| Factors of 18 | 1, 2, 3, 6, 9, 18 |
| Factors of 19 | 1, 19 |
| Factors of 20 | 1, 2, 4, 5, 10, 20 |
| Factors of 21 | 1, 3, 7, 21 |
| Factors of 22 | 1, 2, 11, 22 |
| Factors of 23 | 1, 23 |
| Factors of 24 | 1, 2, 3, 4, 6, 8, 12, 24 |
| Factors of 25 | 1, 5, 25 |
| Factors of 26 | 1, 2, 13, 26 |
| Factors of 27 | 1, 3, 9, 27 |
| Factors of 28 | 1, 2, 4, 7, 14, 28 |
| Factors of 29 | 1, 29 |
| Factors of 30 | 1, 2, 3, 5, 6, 10, 15, 30 |
| Factors of 31 | 1, 31 |
| Factors of 32 | 1, 2, 4, 8, 16, 32 |
| Factors of 33 | 1, 3, 11, 33 |
| Factors of 34 | 1, 2, 17, 34 |
| Factors of 35 | 1, 5, 7, 35 |
| Factors of 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
| Factors of 37 | 1, 37 |
| Factors of 38 | 1, 2, 19, 38 |
| Factors of 39 | 1, 3, 13, 39 |
| Factors of 40 | 1, 2, 4, 5, 8, 10, 20, 40 |
| Factors of 41 | 1, 41 |
| Factors of 42 | 1, 2, 3, 6, 7, 14, 21, 42 |
| Factors of 43 | 1, 43 |
| Factors of 44 | 1, 2, 4, 11, 22, 44 |
| Factors of 45 | 1, 3, 5, 9, 15, 45 |
| Factors of 46 | 1, 2, 23, 46 |
| Factors of 47 | 1, 47 |
| Factors of 48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 |
| Factors of 49 | 1, 7, 49 |
| Factors of 50 | 1, 2, 5, 10, 25, 50 |
| Factors of 51 | 1, 3, 17, 51 |
| Factors of 52 | 1, 2, 4, 13, 26, 52 |
| Factors of 53 | 1, 53 |
| Factors of 54 | 1, 2, 3, 6, 9, 18, 27, 54 |
| Factors of 55 | 1, 5, 11, 55 |
| Factors of 56 | 1, 2, 4, 7, 8, 14, 28, 56 |
| Factors of 57 | 1, 3, 19, 57 |
| Factors of 58 | 1, 2, 29, 58 |
| Factors of 59 | 1, 59 |
| Factors of 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 |
| Factors of 61 | 1, 61 |
| Factors of 62 | 1, 2, 31, 62 |
| Factors of 63 | 1, 3, 7, 9, 21, 63 |
| Factors of 64 | 1, 2, 4, 8, 16, 32, 64 |
| Factors of 65 | 1, 5, 13, 65 |
| Factors of 66 | 1, 2, 3, 6, 11, 22, 33, 66 |
| Factors of 67 | 1, 67 |
| Factors of 68 | 1, 2, 4, 17, 34, 68 |
| Factors of 69 | 1, 3, 23, 69 |
| Factors of 70 | 1, 2, 5, 7, 10, 14, 35, 70 |
| Factors of 71 | 1, 71 |
| Factors of 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 |
| Factors of 73 | 1, 73 |
| Factors of 74 | 1, 2, 37, 74 |
| Factors of 75 | 1, 3, 5, 15, 25, 75 |
| Factors of 76 | 1, 2, 4, 19, 38, 76 |
| Factors of 77 | 1, 7, 11, 77 |
| Factors of 78 | 1, 2, 3, 6, 13, 26, 39, 78 |
| Factors of 79 | 1, 79 |
| Factors of 80 | 1, 2, 4, 5, 8, 10, 16, 20, 40, 80 |
| Factors of 81 | 1, 3, 9, 27, 81 |
| Factors of 82 | 1, 2, 41, 82 |
| Factors of 83 | 1, 83 |
| Factors of 84 | 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 |
| Factors of 85 | 1, 5, 17, 85 |
| Factors of 86 | 1, 2, 43, 86 |
| Factors of 87 | 1, 3, 29, 87 |
| Factors of 88 | 1, 2, 4, 8, 11, 22, 44, 88 |
| Factors of 89 | 1, 89 |
| Factors of 90 | 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 |
| Factors of 91 | 1, 7, 13, 91 |
| Factors of 92 | 1, 2, 4, 23, 46, 92 |
| Factors of 93 | 1, 3, 31, 93 |
| Factors of 94 | 1, 2, 47, 94 |
| Factors of 95 | 1, 5, 19, 95 |
| Factors of 96 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 |
| Factors of 97 | 1, 97 |
| Factors of 98 | 1, 2, 7, 14, 49, 98 |
| Factors of 99 | 1, 3, 9, 11, 33, 99 |
| Factors of 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 |