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Find all common factors of multiple numbers with visualization
Common factors are numbers that divide evenly into two or more numbers. They're the factors that the numbers share. The greatest common factor (GCF) is the largest of these common factors.
Example: Common factors of 12 and 18
1. Always Include 1
1 is a factor of every number, so 1 is always a common factor of any set of numbers.
2. All Divide the GCF
Every common factor divides the GCF. In fact, the set of common factors equals the set of factors of the GCF.
3. Limited by Smallest Number
Common factors cannot be larger than the smallest number in the set.
4. Coprime Numbers
If two numbers share no common factors except 1, they're called coprime or relatively prime (GCF = 1).
If all numbers are the same (e.g., 12, 12, 12), the GCF equals that number, and all its factors are common factors.
If GCF = 1, the numbers are coprime. They share no prime factors. Only common factor is 1.
If one number divides all others (e.g., 6, 12, 18), the GCF equals the smallest number.
For powers of the same prime (e.g., 8, 16, 32), the GCF is the smallest power of that prime.
Common factors are all the numbers that divide evenly into all given numbers, while the GCF (Greatest Common Factor) is specifically the largest of these common factors. For example, 12 and 18 have common factors 1, 2, 3, and 6, but the GCF is 6.
Yes! This is a key relationship: the set of common factors of several numbers is exactly equal to the set of factors of their GCF. So instead of finding all common factors directly, you can find the GCF first, then find all its factors.
No, adding more numbers can only reduce the number of common factors, never increase them. Each additional number adds another constraint. For example, 12 and 18 have common factors {1,2,3,6}, but 12, 18, and 30 only have {1,2,3,6} (same) or fewer.
Method 1: Find GCF of all numbers (using repeated GCF calculations: GCF(a,b,c) = GCF(GCF(a,b),c)), then find all factors of the GCF. Method 2: List all factors of each number, then find the intersection of all lists.
When two numbers have only 1 as a common factor, they're called 'coprime', 'relatively prime', or 'mutually prime'. This means they share no prime factors. For example, 8 and 15 are coprime (8=2³, 15=3×5) even though neither is prime.
To simplify a fraction, divide both numerator and denominator by their GCF. For example, 12/18 simplifies to 2/3 by dividing both by their GCF of 6. You can also divide by any common factor repeatedly: 12/18 → 6/9 (÷2) → 2/3 (÷3).