GCF and LCM Calculator

The GCF (Greatest Common Factor), also called GCD (Greatest Common Divisor) or HCF (Highest Common Factor), is the largest positive integer that divides all the given numbers without leaving a remainder. The LCM (Least Common Multiple) is the smallest positive integer that is divisible by all the given numbers. For example, GCF(12, 18) = 6 and LCM(12, 18) = 36. GCF is used to simplify fractions; LCM is used to find common denominators.

Write each number as a product of prime factors. The GCF is the product of all prime factors that appear in every number, using the lowest exponent that appears. For GCF(24, 36): 24 = 2³ × 3 and 36 = 2² × 3². Both have 2 and 3. Lowest power of 2 is 2² = 4; lowest power of 3 is 3¹ = 3. GCF = 4 × 3 = 12.

Write each number as a product of prime factors. The LCM is the product of all prime factors that appear in any number, using the highest exponent that appears. For LCM(12, 18): 12 = 2² × 3 and 18 = 2 × 3². Highest power of 2 is 2² = 4; highest power of 3 is 3² = 9. LCM = 4 × 9 = 36.

The Euclidean algorithm is an efficient method: divide the larger number by the smaller, take the remainder, then repeat. GCF equals the last non-zero remainder. Example: GCF(48, 18). Step 1: 48 = 2 × 18 + 12. Step 2: 18 = 1 × 12 + 6. Step 3: 12 = 2 × 6 + 0. GCF = 6 (the last non-zero remainder). This works because GCF(a, b) = GCF(b, a mod b).

For any two positive integers a and b: GCF(a, b) × LCM(a, b) = a × b. This is a key identity. It means you only need to find one of GCF or LCM and can derive the other: LCM(a, b) = a × b ÷ GCF(a, b). For example, GCF(4, 6) = 2 and LCM(4, 6) = 12. Check: 2 × 12 = 24 = 4 × 6. Note: this identity applies to exactly two numbers; for three or more, it does not hold.

If GCF(a, b) = 1, the numbers are called coprime or relatively prime. They share no common prime factors. Their LCM equals their product: LCM = a × b. For example, GCF(8, 9) = 1 (8 = 2³, 9 = 3²), so LCM(8, 9) = 72. Any two consecutive integers are always coprime. Any prime number is coprime with any other number that is not its multiple.

To reduce a fraction to lowest terms, divide both numerator and denominator by their GCF. For 36/48: GCF(36, 48) = 12. Divide both by 12: 36/12 = 3, 48/12 = 4. The simplified fraction is 3/4. This works because dividing numerator and denominator by the same non-zero number does not change the value of the fraction.

To add fractions with different denominators, find the LCD (Least Common Denominator), which equals the LCM of the denominators. For 1/4 + 5/6: LCM(4, 6) = 12. Convert: 1/4 = 3/12 and 5/6 = 10/12. Add: 3/12 + 10/12 = 13/12 = 1 1/12. Without the LCM, you would use a larger common denominator (like 24), creating bigger numbers and requiring extra simplification at the end.

Yes. For multiple numbers, compute iteratively. GCF(a, b, c) = GCF(GCF(a, b), c). Similarly, LCM(a, b, c) = LCM(LCM(a, b), c). The prime factorization method also generalises: for GCF, take the lowest power of primes common to all numbers; for LCM, take the highest power of any prime appearing in any number. Our calculator handles 2 to 8 numbers at once.

By convention, GCF(0, n) = n for any positive integer n. This is because every integer divides 0 (0 ÷ n = 0 with no remainder for any non-zero n). Calculators and algorithms typically require positive inputs; our calculator enforces this. The Euclidean algorithm handles it by considering that all positive integers divide zero, making the last non-zero number (n) the GCF.

GCF has several practical applications: (1) Simplifying fractions to lowest terms for cooking or measurement conversions. (2) Dividing objects into equal groups — GCF(36, 48) = 12 means you can make 12 identical groups using 36 and 48 items, with 3 and 4 items respectively. (3) Solving gear ratio and scheduling problems. (4) Cryptography, where coprime numbers are used in RSA encryption algorithms.

LCM appears frequently in scheduling and synchronisation problems. If Event A occurs every 4 days and Event B every 6 days, they next coincide after LCM(4, 6) = 12 days. In music, LCM determines when rhythmic patterns align. In electronics, it helps synchronise signals with different frequencies. In cooking, if one recipe serves 4 and another serves 6, LCM(4, 6) = 12 servings is the minimum to make equal batches of both.

For small numbers, you can list all factors of each number and find the largest one in common. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. Common factors: 1, 2, 3, 6. Greatest = 6. This method works for small numbers but becomes impractical for large ones. The Euclidean algorithm or prime factorization is more efficient for larger values.