Common Factor Calculator
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24, 36 and 60 have 6 common factors: 1, 2, 3, 4, 6, 12
1. Find the greatest common factor (GCF) with Euclid's algorithm: GCF(24, 36): 36 = 1 × 24 + 12; 24 = 2 × 12 + 0 → 12 GCF(12, 60): 60 = 5 × 12 + 0 → 12 GCF = 12 2. A number divides every number in the list exactly when it divides their GCF, so the common factors are the factors of 12. 3. 12 = 2² × 3, which has (2 + 1) × (1 + 1) = 6 factors: 1, 2, 3, 4, 6, 12 Check by listing the factors of each number: 24: 1, 2, 3, 4, 6, 8, 12, 24 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 In every list: 1, 2, 3, 4, 6, 12
| Number | Prime factorization | Number of factors |
|---|---|---|
| 24 | 2³ × 3 | 8 |
| 36 | 2² × 3² | 9 |
| 60 | 2² × 3 × 5 | 12 |
How to use the common factor calculator
- Type two to twenty whole numbers, separated by commas or spaces, for example
24, 36, 60. - Read the list of common factors, their count and the greatest common factor (GCF).
- Open the working to see how the GCF was found and, for numbers up to 10,000, the factor list of each number. The table gives each number's prime factorization.
Common factor formula
If GCF = px × qy × …, the number of common factors = (x + 1)(y + 1) …
A common factor of a list of whole numbers a, b, c is a number that divides every one of them with no remainder. Rather than listing all the factors of each number and comparing the lists, the calculator finds the GCF first with Euclid's algorithm and then lists the factors of that single number, where p and q are the primes in the GCF and x and y their exponents. Both routes give the same answer, but the GCF route stays quick even for numbers with 30 digits.
Worked example: 24, 36 and 60
Step 1: the GCF. GCF(24, 36): 36 = 1 × 24 + 12; 24 = 2 × 12 + 0, so the result is 12. GCF(12, 60): 60 = 5 × 12 + 0, so the result is 12.
Step 2: factor the GCF. 12 = 2² × 3, which has (2 + 1) × (1 + 1) = 6 factors: 1, 2, 3, 4, 6, 12. These are the common factors.
Check by listing. The shared factors are in bold:
- 24: 1, 2, 3, 4, 6, 8, 12, 24
- 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Common factors of familiar pairs
| Numbers | GCF | Common factors | Count |
|---|---|---|---|
| 12, 18 | 6 | 1, 2, 3, 6 | 4 |
| 16, 24 | 8 | 1, 2, 4, 8 | 4 |
| 24, 36 | 12 | 1, 2, 3, 4, 6, 12 | 6 |
| 30, 45 | 15 | 1, 3, 5, 15 | 4 |
| 20, 50 | 10 | 1, 2, 5, 10 | 4 |
| 42, 56 | 14 | 1, 2, 7, 14 | 4 |
| 48, 72 | 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 |
| 60, 90 | 30 | 1, 2, 3, 5, 6, 10, 15, 30 | 8 |
| 100, 150 | 50 | 1, 2, 5, 10, 25, 50 | 6 |
| 9, 16 | 1 | 1 | 1 |
| 36, 48, 60 | 12 | 1, 2, 3, 4, 6, 12 | 6 |
| 360, 840 | 120 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 | 16 |
Where common factors help
- Simplifying fractions step by step. Any common factor can be canceled: 24/36 ÷ 2 = 12/18, then ÷ 6 = 2/3. Dividing by the GCF, 12, does it in one step.
- Equal groups. With 24 red and 36 blue counters, the possible numbers of identical groups with nothing left over are the common factors: 1, 2, 3, 4, 6 or 12 groups.
- Tiles and grids. A 24 × 36 rectangle can be covered by whole squares of side 1, 2, 3, 4, 6 or 12 with no cutting.
Tips
- 1 is always a common factor, so the list is never empty. If it is the only one, the numbers are coprime.
- The first common factor after 1 is always prime. The smallest common factor above 1 can't have a smaller factor of its own, or that factor would come first.
- Adding a number can only shorten the list. Every extra number is one more test each common factor must pass.
- For the largest common factor alone, with three methods worked out, use the GCF calculator. For every factor of one number, use the factor calculator.
Frequently asked questions
What are the common factors of 24 and 36?
They are 1, 2, 3, 4, 6 and 12. The greatest common factor of 24 and 36 is 12, and the common factors are exactly the six factors of 12.
How are common factors different from the GCF?
The GCF is a single number, the largest factor the numbers share. The common factors are all the shared factors, from 1 up to the GCF. For 24, 36 and 60 the GCF is 12, while the common factors are 1, 2, 3, 4, 6 and 12.
Why are the common factors the same as the factors of the GCF?
Every factor of the GCF divides each number, because the GCF does. In the other direction, any number that divides all of them also divides their GCF (Euclid’s algorithm only ever subtracts multiples, which keeps every shared factor). So the two lists are the same.
Do any two numbers always have a common factor?
Yes: 1 divides every whole number. When 1 is the only common factor, the numbers are called relatively prime or coprime. 9 and 16 are coprime even though neither of them is prime.
How many common factors do 360 and 840 have?
Sixteen. Their GCF is 120 = 2³ × 3 × 5, which has (3 + 1) × (1 + 1) × (1 + 1) = 16 factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120.