Factors Calculator
Factors - Solve mathematical problems with step-by-step solutions.
Factors Calculator
Find all factors of a number
What are Factors?
Factors of a number are integers that can be multiplied together to get that number. In other words, a factor is a number that divides another number evenly, without leaving a remainder.
Example: The factors of 12 are 1, 2, 3, 4, 6, and 12.
How the Factors Calculator Works
Factors are whole numbers that divide evenly into another number without leaving a remainder. In other words, if you can multiply two whole numbers together to get a target number, those two numbers are factors of the target. Understanding factors is fundamental to many areas of mathematics, including simplifying fractions, finding common denominators, and solving algebraic equations.
What Are Factors?
For any given number, a factor is a number that divides into it exactly. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because:
- 1 × 12 = 12
- 2 × 6 = 12
- 3 × 4 = 12
Notice that factors come in pairs. Every number has at least two factors: 1 and itself. Numbers with exactly two factors are called prime numbers, while numbers with more than two factors are called composite numbers.
How to Find Factors
To find all factors of a number systematically:
- Start with 1 (always a factor of every number)
- Test each whole number in order to see if it divides evenly into your target number
- When you find a factor, note both it and its pair (the number you multiply it by to get the target)
- Continue until you reach the square root of the number - beyond this point, you'll only find pairs you've already discovered
- List all factors in ascending order
Prime vs. Composite Numbers
- Prime numbers: Have exactly two factors (1 and themselves). Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
- Composite numbers: Have more than two factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18...
- Special case: The number 1 is neither prime nor composite - it has only one factor (itself)
Prime Factorization
Every composite number can be expressed as a unique product of prime numbers, called its prime factorization. For example, 24 = 2 × 2 × 2 × 3 = 23 × 3. Prime factorization is useful for finding GCF, LCM, and simplifying fractions.
Factor Finding Examples
Example 1: Factors of 24
Find all factors of 24:
Method: Test each number from 1 up to 24
• 24 ÷ 1 = 24 ✓ (factors: 1 and 24)
• 24 ÷ 2 = 12 ✓ (factors: 2 and 12)
• 24 ÷ 3 = 8 ✓ (factors: 3 and 8)
• 24 ÷ 4 = 6 ✓ (factors: 4 and 6)
• 24 ÷ 5 = 4.8 ✗ (not a whole number)
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 (eight factors total)
Example 2: Factors of a Prime Number (17)
Find all factors of 17:
Testing: 17 is only divisible by 1 and 17
• 17 ÷ 1 = 17 ✓
• 17 ÷ 2 = 8.5 ✗
• 17 ÷ 3 = 5.67... ✗
• (Continue testing - none divide evenly)
Factors of 17: 1, 17 (only two factors - this is a prime number)
Example 3: Factors of 36
Find all factors of 36:
Factor pairs:
• 1 × 36 = 36
• 2 × 18 = 36
• 3 × 12 = 36
• 4 × 9 = 36
• 6 × 6 = 36 (6 appears only once in the list)
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 (nine factors total)
36 is called a perfect square because one of its factors (6) multiplied by itself gives 36.
Example 4: Prime Factorization of 60
Express 60 as a product of prime factors:
Method: Divide by the smallest prime numbers repeatedly
• 60 ÷ 2 = 30
• 30 ÷ 2 = 15
• 15 ÷ 3 = 5
• 5 is prime, so we stop
Prime factorization: 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
All factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Tips for Working with Factors
Stop at the Square Root
When finding factors, you only need to test numbers up to the square root of your target number. After that point, you're just finding the pairs of factors you've already discovered. For example, when finding factors of 100, you only need to test up to 10 (since √100 = 10). This saves significant time with larger numbers.
Check Divisibility Rules
Learn quick divisibility rules to speed up factor finding: A number is divisible by 2 if it's even; by 3 if the sum of its digits is divisible by 3; by 5 if it ends in 0 or 5; by 10 if it ends in 0. These shortcuts help you quickly identify factors without full division.
List Factors in Pairs
When finding factors, work with pairs to ensure you don't miss any. If 3 is a factor of 24, then 8 must also be a factor (since 3 × 8 = 24). This pairing approach helps you systematically find all factors without overlooking any.
Use Prime Factorization for Complex Numbers
For larger numbers, finding the prime factorization first can help you systematically determine all factors. Once you have the prime factorization, you can combine the prime factors in different ways to find all possible factors. For example, if 72 = 23 × 32, you can create factors by using different combinations of these primes.
Remember 1 and the Number Itself
Every whole number greater than zero has at least two factors: 1 and the number itself. Always start your factor list with 1 and end with the number you're factoring. The only exception is 1 itself, which has only one factor: 1.
Frequently Asked Questions
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