Factor Calculator
Find all factors of any number with DigiCalc's free factor calculator. Prime factorization, factor pairs, and divisibility rules included.
A factor calculator finds every whole number that divides evenly into a given number without leaving a remainder. Whether you are simplifying fractions, identifying prime factors of a number, or solving number theory problems, DigiCalc's factor calculator delivers accurate results instantly. This factor finder works for any positive integer and returns all factors, factor pairs, and a complete prime factorization breakdown. Understanding math factors is a core skill in arithmetic, algebra, and beyond.
What Is a Factor?
A factor is any whole number that divides exactly into another number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each divides into 12 evenly. Every whole number has at least two factors: 1 and itself. Numbers with exactly two factors are called prime numbers. Numbers with more than two factors are called composite numbers. The number 1 is neither prime nor composite.
How to Find Factors of a Number
To find all factors of a number, divide it by every whole number from 1 up to its square root. When a division produces no remainder, both the divisor and its result are factors. This method is systematic and guarantees no factor is missed.
Example: factors of 36
- 36 / 1 = 36 — factors: 1 and 36
- 36 / 2 = 18 — factors: 2 and 18
- 36 / 3 = 12 — factors: 3 and 12
- 36 / 4 = 9 — factors: 4 and 9
- 36 / 6 = 6 — factor: 6 (square root, paired with itself)
All factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The total count of factors — called the divisor count — is determined by the exponents in the prime factorization. For 36 = 2² x 3², the divisor count is (2+1) x (2+1) = 9 factors.
Prime Factorization Calculator — What It Shows
Prime factorization breaks a number into its prime number building blocks. Every composite number has exactly one unique set of prime factors. DigiCalc's prime factorization calculator displays this breakdown automatically alongside the full factor list.
Example: prime factorization of 60
60 = 2 x 2 x 3 x 5 = 2² x 3 x 5
The prime factors of 60 are 2, 3, and 5. Prime factorization is used in cryptography, simplifying fractions, and computing the greatest common factor (GCF) or least common multiple (LCM) of two numbers. A factoring calculator with steps shows each division in sequence, making it easy to verify the work. According to the NIST Digital Library of Mathematical Functions, prime factorization is foundational to modern number theory. It also underlies RSA encryption — the security standard used by every secure website and online banking platform worldwide.
Factor Pairs
Factor pairs are two numbers that multiply together to produce the original number. Every factor below the square root has a corresponding pair above it.
| Number | Factor Pairs | Total Factors |
|---|---|---|
| 12 | (1, 12), (2, 6), (3, 4) | 6 |
| 20 | (1, 20), (2, 10), (4, 5) | 6 |
| 36 | (1, 36), (2, 18), (3, 12), (4, 9), (6, 6) | 9 |
| 48 | (1, 48), (2, 24), (3, 16), (4, 12), (6, 8) | 10 |
| 100 | (1, 100), (2, 50), (4, 25), (5, 20), (10, 10) | 9 |
Factors of 42
The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. The number 42 has 8 factors in total. Its prime factorization is 2 x 3 x 7. Factor pairs of 42 are (1, 42), (2, 21), (3, 14), and (6, 7). Since 42 has more than two factors, it is a composite number. The GCF of 42 and 28, for example, is 14, making the fraction 42/28 reducible to 3/2.
Factors of 45
The factors of 45 are 1, 3, 5, 9, 15, and 45. The number 45 has 6 factors. Its prime factorization is 3² x 5. Factor pairs of 45 are (1, 45), (3, 15), and (5, 9). The GCF of 45 and 30 is 15, which means the fraction 30/45 simplifies to 2/3. Because 45 ends in 5, it is always divisible by 5.
Factors of 15
The factors of 15 are 1, 3, 5, and 15. The number 15 has 4 factors and its prime factorization is 3 x 5. Factor pairs of 15 are (1, 15) and (3, 5). Although 15 may appear prime, it is composite: its digit sum (1 + 5 = 6) is divisible by 3, confirming divisibility. It also ends in 5, confirming divisibility by 5.
Factors of 75
The factors of 75 are 1, 3, 5, 15, 25, and 75. The number 75 has 6 factors. Its prime factorization is 3 x 5². Factor pairs of 75 are (1, 75), (3, 25), and (5, 15). Because 75 equals three-quarters of 100, its factors are practical for dividing quantities into equal groups. For example, 75 students divided into groups of 25 gives exactly 3 equal groups with no remainder.
Factors of 57
The factors of 57 are 1, 3, 19, and 57. The number 57 has only 4 factors, and its prime factorization is 3 x 19. Factor pairs of 57 are (1, 57) and (3, 19). Despite looking prime at first glance, 57 is composite. Its digit sum (5 + 7 = 12) is divisible by 3, which confirms 3 is a factor. Dividing 57 by 3 gives 19, which is itself a prime number.
Factors of 85
The factors of 85 are 1, 5, 17, and 85. The number 85 has 4 factors. Its prime factorization is 5 x 17. Factor pairs of 85 are (1, 85) and (5, 17). Because 85 ends in 5, it is divisible by 5. Dividing 85 by 5 gives 17, which is prime. So 85 has exactly two prime factors: 5 and 17.
Factors and Multiples — What Is the Difference?
Factors and multiples are related but opposite ideas. A factor divides into a number. A multiple is the result of multiplying a number. Factors are always finite for any given number. Multiples are always infinite.
| Concept | Definition | Example (for 6) |
|---|---|---|
| Factors of 6 | Numbers that divide 6 evenly | 1, 2, 3, 6 |
| Multiples of 6 | Results of multiplying 6 by whole numbers | 6, 12, 18, 24, 30... |
A common confusion: 12 is a multiple of 6 but also a factor of 24. The relationship always depends on the reference number. Factors are also called divisors — both terms refer to numbers that divide evenly into a given value. To find the greatest common factor of two numbers, list all divisors of each, then identify the largest one they share.
There are infinitely many prime numbers. The largest known prime number, discovered in 2024, contains over 41 million digits. Every composite number can be expressed as a unique product of prime factors. This principle is the Fundamental Theorem of Arithmetic.
Divisibility Rules for Finding Factors Quickly
These rules let you identify factors without performing full long division:
| Divisible by | Rule | Quick example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 84: ends in 4 — yes |
| 3 | Sum of digits is divisible by 3 | 57: 5+7=12 — yes |
| 4 | Last two digits divisible by 4 | 312: 12/4=3 — yes |
| 5 | Last digit is 0 or 5 | 85: ends in 5 — yes |
| 6 | Divisible by both 2 and 3 | 42: even and digit sum 6 — yes |
| 9 | Sum of digits divisible by 9 | 81: 8+1=9 — yes |
| 10 | Last digit is 0 | 90: ends in 0 — yes |
How to Factor Completely
To factor completely means to break a number down until every component is a prime number and no further factoring is possible. For whole numbers, factoring completely is identical to finding the prime factorization.
Example: factor completely 72
72 = 2 x 36. Then 36 = 2 x 18. Then 18 = 2 x 9. Then 9 = 3 x 3. Result: 72 = 2³ x 3²
Every component (2 and 3) is now prime. The number 72 is fully factored. A factor completely calculator automates this step-by-step process and displays the full factor tree, making it easy to verify each step.
Real-World Uses of Factors
Factors appear in practical situations across many fields:
- Simplifying fractions: To reduce 18/24, use a find the greatest common factor calculator to get the GCF (6), then divide both: 18/24 = 3/4.
- Dividing into equal groups: 48 items have 10 possible group sizes. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
- Scheduling and planning: If one task repeats every 6 days and another every 9 days, the LCM (18) shows when both tasks coincide. Finding the LCM requires factoring both numbers first.
- Cryptography: RSA encryption, which protects every secure website and bank transaction, relies on the difficulty of factoring very large numbers. A 300-digit number would take millions of years to factor with current computing technology.
- Architecture and tiling: Tiling a 45 by 75 rectangle requires knowing the GCF (15). That gives the largest square tile that fits without cutting.
Limitations of This Factor Calculator
DigiCalc's factor calculator works accurately for standard positive integers. There are constraints to be aware of:
- Whole numbers only: The calculator does not support decimals or fractions as inputs.
- Positive integers only: Negative numbers and zero are not valid inputs for factoring.
- Polynomial factoring: This tool factors numbers, not algebraic expressions such as x² + 5x + 6. A factor by grouping calculator is needed for multi-term polynomial expressions.
- Very large numbers: Numbers with many digits may take longer to process since prime factorization becomes computationally intensive as numbers grow.
For algebraic or polynomial factoring, a dedicated algebra calculator is required. For related number operations, try DigiCalc's factorial calculator. Explore more tools in our complete math calculators collection.
For more math tools, try DigiCalc's absolute value calculator for signed number operations.