Factor Calculator — Free Online Factor Finder

How to Use the Factor Calculator

1. Enter a positive integer: Type any number from 1 to 1,000,000 into the input field. The default value is 120, which demonstrates all features since it has many factors.
2. Check the number type: Below the input, the calculator instantly tells you whether your number is prime (exactly 2 factors), composite (more than 2 factors), or neither (the number 1 has only one factor).
3. View the factor count and list: The results panel shows how many factors the number has and lists them all in ascending order. For example, 120 has 16 factors.
4. Examine factor pairs: The factor pairs section shows every pair of numbers whose product equals your input. This is useful for factoring problems in algebra and finding rectangle dimensions in geometry.
5. Read the prime factorization: The bottom section shows the prime factorization — how your number breaks down into a product of prime numbers. This is the most fundamental representation in number theory.

All results update in real time as you change the input number, so you can quickly explore how factor structure changes across different numbers.

Factor Finding Formulas

Variables Explained

• n: The positive integer whose factors you want to find. Every factor f satisfies n mod f = 0 (f divides n exactly).
• p1, p2, ...: The distinct prime factors of n. Each prime appears with an exponent showing how many times it divides n.
• a1, a2, ...: The exponents of each prime factor. The factor count formula multiplies (each exponent + 1) together.
• sqrt(n): The square root of n. When finding factors by trial division, you only need to test up to this value because factor pairs straddle the square root.

Step-by-Step Example

Find all factors of 180:

1. sqrt(180) is approximately 13.4, so test divisors 1 through 13
2. 180/1 = 180 (pair: 1, 180)
3. 180/2 = 90 (pair: 2, 90)
4. 180/3 = 60 (pair: 3, 60)
5. 180/4 = 45 (pair: 4, 45)
6. 180/5 = 36 (pair: 5, 36)
7. 180/6 = 30 (pair: 6, 30)
8. 7, 8, 11, 13 do not divide 180 evenly
9. 180/9 = 20 (pair: 9, 20)
10. 180/10 = 18 (pair: 10, 18)
11. 180/12 = 15 (pair: 12, 15)

All 18 factors: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180. The prime factorization 180 = 2^2 x 3^2 x 5 confirms the count: (2+1)(2+1)(1+1) = 18 factors.

Practical Examples

Example 1: Alex's Classroom Seating

Alex is a teacher arranging 30 desks into equal rows for a classroom. He wants to know all possible rectangular arrangements.

• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
• Factor pairs: (1, 30), (2, 15), (3, 10), (5, 6)
• Possible layouts: 1x30, 2x15, 3x10, 5x6

Alex has four possible rectangular arrangements. The 5x6 layout is likely the best for a classroom, being closest to a square shape. Factor pairs directly correspond to rectangle dimensions when the area is fixed — a principle used in architecture, gardening, and interior design.

Example 2: Mia's Cookie Distribution

Mia baked 84 cookies and wants to divide them equally among party guests, with each guest getting more than one cookie. She needs to know which group sizes work.

• Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
• Valid group sizes (more than 1 cookie each): 2, 3, 4, 6, 7, 12, 14, 21, 28, 42
• For 7 guests: 84/7 = 12 cookies each

Mia can choose from 10 different group sizes where each guest gets the same number of cookies (more than one). Factors tell you exactly which even distributions are possible, making them essential for fair division problems in everyday life.

Example 3: Jason's Algebraic Factoring

Jason needs to factor the quadratic expression x^2 + 11x + 24. He needs two numbers that multiply to 24 and add to 11.

• Factor pairs of 24: (1, 24), (2, 12), (3, 8), (4, 6)
• Check sums: 1+24=25, 2+12=14, 3+8=11, 4+6=10
• The pair (3, 8) works: 3+8=11 and 3x8=24
• Result: x^2 + 11x + 24 = (x + 3)(x + 8)

By listing the factor pairs of 24, Jason quickly identified which pair sums to the linear coefficient. This is the standard method for factoring trinomials in algebra, and having a complete list of factor pairs makes the process systematic rather than guess-and-check.

Number of Factors Reference Table

Tips and Complete Guide

Efficient Factor Finding

You never need to test divisors beyond the square root of a number. If d is a factor of n where d is less than or equal to sqrt(n), then n/d is also a factor that is greater than or equal to sqrt(n). This means every factor less than or equal to the square root has a partner above it. For the number 100, you only test 1 through 10, finding (1,100), (2,50), (4,25), (5,20), (10,10). This square root optimization is used in professional factoring algorithms and reduces the work dramatically for large numbers.

Highly Composite Numbers

Some numbers have exceptionally many factors relative to their size. These are called highly composite numbers: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 1260, and so on. The number 12 is highly composite because it has 6 factors — more than any smaller number. These numbers appear naturally in measurement systems: there are 12 inches in a foot and 360 degrees in a circle precisely because these highly composite numbers allow for many even divisions.

Factors in Algebra

Finding factors of the constant term is crucial when factoring polynomials. For a quadratic ax^2 + bx + c, you need factor pairs of a x c that sum to b. For cubics, the Rational Root Theorem states that any rational root p/q has p as a factor of the constant term and q as a factor of the leading coefficient. Having a complete list of factors makes these algebraic techniques systematic.

Perfect Numbers and Factor Sums

When you add all factors of a number excluding the number itself, interesting patterns emerge. If the sum equals the number, it is called a perfect number. The smallest perfect numbers are 6 (1+2+3=6), 28 (1+2+4+7+14=28), and 496. If the sum is less than the number, it is deficient; if greater, it is abundant. For example, 12 is abundant because 1+2+3+4+6=16 > 12. These classifications have been studied since ancient Greece and connect to open problems in modern number theory.

Common Mistakes to Avoid

• Forgetting 1 and the number itself: Every positive integer has 1 and itself as factors. The number 1 is always the smallest factor and the number itself is always the largest.
• Stopping before the square root: When finding factors manually, test every integer up to and including the floor of the square root. Missing even one divisor means missing its partner too.
• Confusing factors with multiples: Factors divide into a number (they are smaller or equal); multiples are products of a number (they are larger or equal, starting from the number itself).
• Thinking 1 is prime: The number 1 is neither prime nor composite. It has exactly one factor (itself), while primes have exactly two and composites have three or more.
• Double-counting the square root: For perfect squares like 36, sqrt(36) = 6 appears as a factor only once (not twice), even though the pair is (6, 6). Factor counting must avoid this duplication.

Frequently Asked Questions