How it works
The GCF (greatest common factor) is the largest positive integer that divides evenly into all the numbers you're checking. It's also known as the GCD (greatest common divisor) or HCF (highest common factor)—these terms are interchangeable.
When you enter two or more numbers, the calculator finds all the factors of each one, then identifies which factors appear in every number. The largest of those shared factors is your answer.
For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The common factors are 1, 2, 3, 4, 6, and 12—so the GCF is 12.
The formula
GCF = the largest positive integer that divides all given numbers with remainder 0
The calculator typically uses the Euclidean algorithm, which is fast and efficient. For two numbers a and b, it repeatedly replaces the larger number with the remainder of dividing the larger by the smaller, until the remainder is zero. The last non-zero remainder is the GCF.
For more than two numbers, the calculator finds the GCF of the first two, then finds the GCF of that result and the third number, and so on.
Worked example
Let's find the GCF of 48, 64, and 80.
Step 1: Start with the first two numbers (48 and 64).
- 64 ÷ 48 = 1 remainder 16
- 48 ÷ 16 = 3 remainder 0
- The GCF of 48 and 64 is 16.
Step 2: Now find the GCF of that result (16) and the third number (80).
- 80 ÷ 16 = 5 remainder 0
- The GCF of 16 and 80 is 16.
Result: The GCF of 48, 64, and 80 is 16.
You can verify: 48 = 16 × 3, 64 = 16 × 4, and 80 = 16 × 5. Since 3, 4, and 5 share no common factors, 16 is indeed the greatest.
Common mistakes
Confusing GCF with LCM: The least common multiple (LCM) is the smallest number that all your numbers divide into evenly—the opposite of GCF. For 12 and 18, the GCF is 6, but the LCM is 36.
Forgetting to check all factors: Don't stop at the first common factor you find. You need the largest one. When simplifying a fraction like 24/36, using GCF = 12 gives you 2/3 in one step, rather than reducing multiple times.
Assuming GCF is always greater than 1: If your numbers are coprime (like 7 and 11, or 9 and 16), the GCF is 1. This is a valid result, not an error.
Ignoring order or repetition: GCF doesn't care about the order you enter numbers or whether you repeat one. The GCF of (12, 18) is the same as (18, 12), and the GCF of (12, 18, 12) is also 6.