Factor Calculator: Find All Factors, Prime Factors And Factored Form

Factoring shows up everywhere in math, from simplifying fractions to solving equations. This guide covers factors of numbers, prime factorization and algebra factoring. Every example was checked with a computer algebra system.

Note: Calc Solver (calc-solver.co) is an independent guide site and is not affiliated with any calculator mentioned here.

What A Factor Is

A factor is a whole number that divides into another number with no remainder. 4 is a factor of 12 because 12 ÷ 4 = 3 exactly. 5 is not, because 12 ÷ 5 leaves a remainder of 2.

Two factors that multiply to the number form a factor pair. The factor pairs of 12 are 1 × 12, 2 × 6 and 3 × 4. "Divisor" means the same as factor, while "multiple" goes the other way: 12 is a multiple of 4.

How To Find All Factors Of A Number

To find all factors, divide the number by 1, 2, 3 and so on, up to its square root. Each divisor that works gives you a factor pair. This method is called trial division, and it is what a factor calculator does in the background.

Here is 36. Its square root is 6, so you only test 1 through 6:

So the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

You can stop at the square root because every pair has one factor below it and one above it. Notice that 36 has an odd number of factors, 9. Only perfect squares do that, because one pair repeats (6 × 6). For a negative number like −12, the factors are the same numbers with both signs: ±1, ±2, ±3, ±4, ±6 and ±12.

Divisibility Rules That Speed Up Factoring

Divisibility rules let you test a number in your head before you divide, so you skip divisions that cannot work.

Prime Numbers Vs Composite Numbers

A prime number has exactly two factors: 1 and itself. A composite number has more than two. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

Three facts trip people up:

• 1 is neither prime nor composite. It has only one factor.
• 2 is the only even prime. Every other even number has 2 as a factor.
• Odd does not mean prime. 91 looks prime, but 91 = 7 × 13.

To test a number, divide by each prime up to its square root. For 97, test 2, 3, 5 and 7. None work, so 97 is prime.

Prime Factorization Calculator: How It Works

A prime factorization calculator writes a number as a product of primes only, like 360 = 2³ × 3² × 5. Every whole number greater than 1 has exactly one prime factorization, apart from the order. This fact is called the fundamental theorem of arithmetic, and Math is Fun's prime factorization page has more practice examples.

Two hand methods give the same answer.

The Division Method (Ladder)

Divide by the smallest prime that works, write the result below, and repeat until you reach 1. Here is 360:

The primes on the left are 2, 2, 2, 3, 3 and 5, so 360 = 2³ × 3² × 5.

The Factor Tree Method

A factor tree splits the number into any factor pair, then keeps splitting until every branch ends in a prime. For 84, you could start with 84 = 4 × 21. Then 4 = 2 × 2 and 21 = 3 × 7. The prime leaves are 2, 2, 3 and 7, so 84 = 2² × 3 × 7.

Starting with 84 = 2 × 42 or 7 × 12 changes the tree, but never the primes at the end.

How To Count Factors Without Listing Them

You can count a number's factors straight from its prime factorization. Add 1 to each exponent, then multiply the results. This saves a lot of time on large numbers.

It works because each factor of 360 uses 0 to 3 twos, 0 to 2 threes and 0 or 1 five: 4 × 3 × 2 = 24 choices.

Using Prime Factors To Find The GCF And LCM

Prime factorization is the most reliable way to find the greatest common factor (GCF) and least common multiple (LCM).

Take 84 = 2² × 3 × 7 and 360 = 2³ × 3² × 5.

• GCF: take each prime the numbers share, at its lowest power. They share 2 and 3, so the GCF is 2² × 3 = 12.
• LCM: take every prime that appears, at its highest power. That is 2³ × 3² × 5 × 7 = 2,520.

Check: GCF × LCM always equals the two numbers multiplied together. 12 × 2,520 = 30,240, and 84 × 360 = 30,240. It matches.

The GCF simplifies fractions (84/360 = 7/30), and the LCM gives common denominators. The fraction calculator does both for you.

Factoring Algebraic Expressions

Factoring a polynomial means rewriting it as a product of simpler expressions. It is the reverse of multiplying out, and it is the main step in solving many equations. Math is Fun's factoring guide covers the basics. Always follow this order:

1. Take out the greatest common factor.
2. Count the terms: two terms suggest a special pattern, three suggest a trinomial, four suggest grouping.
3. Check whether any factor can be factored again.

Taking Out The Greatest Common Factor

The GCF of an expression is the largest number and power of x that divides every term. In 6x³ + 9x², every term is divisible by 3 and by x². So 6x³ + 9x² = 3x²(2x + 3).

Skipping this step often leaves an answer incomplete: 2x² − 8 should become 2(x − 2)(x + 2).

Factoring Trinomials When A = 1

For x² + bx + c, find two numbers that multiply to c and add to b.

• x² + 7x + 12: 3 and 4 multiply to 12 and add to 7, so the answer is (x + 3)(x + 4).
• x² − 5x − 14: −7 and 2 multiply to −14 and add to −5, so the answer is (x − 7)(x + 2).

Factoring Trinomials When A Is Not 1 (AC Method)

For ax² + bx + c, multiply a × c, find two numbers that multiply to ac and add to b, then split the middle term and group.

Take 6x² + 7x − 3:

1. a × c = 6 × (−3) = −18.
2. 9 and −2 multiply to −18 and add to 7.
3. Split the middle: 6x² + 9x − 2x − 3.
4. Group: 3x(2x + 3) − 1(2x + 3).
5. Factor out the shared part: (2x + 3)(3x − 1).

Special Factoring Patterns

Some expressions match a pattern you can factor in one step.

A sum of squares, like x² + 4, does not factor over the real numbers. That is why x⁴ − 16 stops at (x − 2)(x + 2)(x² + 4).

Factoring By Grouping

Use grouping for four terms: factor each pair, then pull out the shared factor.

For x³ + 2x² + 3x + 6: group as (x³ + 2x²) + (3x + 6). That gives x²(x + 2) + 3(x + 2), which becomes (x + 2)(x² + 3).

When An Expression Will Not Factor

Polynomials that cannot be factored using whole numbers are called prime polynomials. For a quadratic, check the discriminant, b² − 4ac. If it is not a perfect square, it does not factor over the integers.

For x² + x + 1, the discriminant is 1 − 4 = −3, so it is prime. To solve an equation like that, use the quadratic formula calculator instead.

Common Factoring Mistakes

Most errors come from stopping too early or mixing up signs.

Summary

• A factor divides a number with no remainder. Test up to the square root to find every factor pair.
• Prime factorization writes a number as a product of primes, like 360 = 2³ × 3² × 5. The ladder and the factor tree give the same result.
• Prime factors also count factors, (a + 1)(b + 1)…, and give the GCF and LCM.
• For expressions, take out the GCF first, then use the trinomial, AC, pattern or grouping method.

Next step: type your number or expression into the factor calculator at the top of this page. To solve equations once they are factored, try the equation solver, or explore more tools on the Calc Solver homepage.

Frequently Asked Questions