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What Is a Factor in Math?
In arithmetic and algebra, a factor of a number is an integer that divides another number evenly, leaving a remainder of zero. For example, because 3 multiplied by 4 equals 12, both 3 and 4 are factors of 12. Factoring is a core algebraic skill used to simplify equations, find common denominators, and solve quadratic trinomials.
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Understanding Factor Pairs
Factors always occur in pairs. A factor pair consists of two numbers that, when multiplied together, equal the target number.
For example, let's find all the factor pairs of the number 24:
- 1 × 24 = 24 → Pair: (1, 24)
- 2 × 12 = 24 → Pair: (2, 12)
- 3 × 8 = 24 → Pair: (3, 8)
- 4 × 6 = 24 → Pair: (4, 6)
Therefore, the complete list of factors for 24 is: 1, 2, 3, 4, 6, 8, 12, and 24. Notice that as the numbers on the left increase, they meet in the middle (between 4 and 6), signaling that we have found all possible factors.
Factors vs. Multiples Comparison
It is common to confuse factors and multiples. Here is a clear comparison table to help you distinguish between the two:
| Feature | Factors | Multiples |
|---|---|---|
| Definition | Numbers that divide evenly into a number. | Numbers you get by multiplying a number by integers. |
| Size Relative to Number | Equal to or smaller than the number. | Equal to or larger than the number. |
| Quantity | Finite (there is a limited set of divisors). | Infinite (multiples go on forever). |
| Example (for 6) | 1, 2, 3, 6 | 6, 12, 18, 24, 30... |
Prime Factorization & Trees
Every composite number can be written as a unique product of prime numbers. This is called the number's prime factorization. You can find this by creating a factor tree:
- Write down the target number (e.g., 36).
- Choose any two factors of that number to branch out (e.g., 6 and 6, or 4 and 9).
- For each branch, if the number is composite, branch it again. If it is prime, circle it.
- Continue branching until only circled prime numbers remain at the bottom of the tree.
Factor Tree Example: 36
· Branch 36 into 6 and 6.
· Branch the first 6 into 2 (prime) and 3 (prime).
· Branch the second 6 into 2 (prime) and 3 (prime).
Combine the prime factors: 2 × 2 × 3 × 3
Write with exponents: 2² × 3² = 36
Factors in Algebra
In algebra, factoring involves rewriting an expression as a product of simpler polynomials. For example, factoring the expression x² + 5x + 6 requires finding two numbers that multiply to 6 and add to 5.
Those numbers are 2 and 3. Therefore, the factored form of the quadratic is:
Factor and Algebra Calculators
Instantly find divisors, GCFs, and solve polynomial factorization with our free math calculators:
Frequently asked questions
A factor is a number that divides into another number evenly (with no remainder). For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. A multiple is the result of multiplying that number by an integer. For example, the multiples of 12 are 12, 24, 36, 48, etc. Factors are smaller than or equal to the number, while multiples are equal to or larger.
Yes, factors can be negative. Since multiplying two negative numbers results in a positive number, negative numbers can be factors. For example, the factor pairs of 6 are (1, 6), (2, 3), (−1, −6), and (−2, −3). In elementary school, only positive factors are usually listed, but algebraically, negative factors are equally valid.
A prime factor is a factor of a number that is itself a prime number. For example, the factors of 20 are 1, 2, 4, 5, 10, and 20. Out of these, the numbers 2 and 5 are prime, so they are the prime factors of 20.
The Greatest Common Factor (GCF or GCD) is the largest integer that divides two or more numbers evenly. For example, the factors of 12 are 1, 2, 3, 4, 6, 12, and the factors of 18 are 1, 2, 3, 6, 9, 18. The shared (common) factors are 1, 2, 3, and 6. The largest of these is 6, so GCF(12, 18) = 6.