Common Factor Calculator - GCD, HCF, and LCM

A common factor calculator takes two or more positive integers and answers three related questions at once: what is the largest integer that divides every input (the greatest common divisor, written GCD and also called the highest common factor or HCF); what is the smallest positive integer that every input divides (the least common multiple, or LCM); and what is the complete list of positive integers that divide all of the inputs. That last list is not a mystery - it is exactly the set of positive divisors of the GCD.

School worksheets often ask you to “list the common factors of 48 and 18” or “find the HCF of 48, 18, and 30.” Those are the same task. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The factors of 18 are 1, 2, 3, 6, 9, 18. The overlap is 1, 2, 3, 6, and the largest of those is 6, which is gcd(48, 18). Once you know the GCD, you do not need to factor every number separately: every common factor divides 6, so the common-factor list is just 1, 2, 3, 6. This page uses that identity so three, four, or twenty numbers are handled the same way.

LCM shows up in a different kind of problem: when will two repeating events coincide, or what denominator should you use to add fractions. For two positives, lcm(a, b) = |a · b| / gcd(a, b). For 48 and 18 that is (48 × 18) / 6 = 144. For more than two numbers you fold: lcm(a, b, c) = lcm(lcm(a, b), c). This free online GCD and LCM tool runs entirely in the browser with BigInt arithmetic, so typical homework sizes and moderately large integers are exact - not floating-point approximations.

The HCF calculator is built for paste-and-read homework checks, not for a multi-step form:

1. Type two or more positive integers in the input box. Separate them with commas, spaces, or new lines. A typical triple is 48, 18, 30.
2. Read the GCD (HCF) in the result panel. That value is the largest integer that divides every number you entered. Euclid’s algorithm is applied pairwise across the whole list.
3. Read the LCM and the common-factor list. The list is every positive divisor of the GCD, printed in ascending order. When the GCD itself is larger than one billion, listing is skipped so the browser stays responsive; GCD and LCM still appear.
4. Optionally copy the full summary, or clear the box to start over. The last input is stored locally in your browser for up to 30 days.

Note: decimals, zero, and negatives are rejected on purpose. GCD and LCM here are defined for positive integers. If you have 1.5 and 2.5, scale them first (15 and 25) and interpret the result in the scaled units.

Euclid’s algorithm replaces a pair (a, b) with (b, a mod b) until the remainder is 0. The last non-zero remainder is the GCD. It never needs a complete prime factorization, which is why it stays fast even when the integers have many digits. This calculator uses BigInt so the remainders are exact.

Example - gcd(48, 18):

• 48 = 2 × 18 + 12, so gcd(48, 18) = gcd(18, 12)
• 18 = 1 × 12 + 6, so gcd(18, 12) = gcd(12, 6)
• 12 = 2 × 6 + 0, so gcd(12, 6) = 6
• GCD / HCF = 6. Common factors: 1, 2, 3, 6.

Example - three numbers 48, 18, 30:

• gcd(48, 18) = 6, then gcd(6, 30) = 6
• lcm(48, 18) = 48 × 18 / 6 = 144, then lcm(144, 30) = 144 × 30 / gcd(144, 30). gcd(144, 30) = 6, so LCM = 720.
• GCD = 6, LCM = 720, common factors still 1, 2, 3, 6 because they are the divisors of 6.

Example - coprime pair 8 and 15:

• gcd(8, 15) = 1 (the numbers are coprime). The only common factor is 1.
• lcm(8, 15) = 120, which is simply the product.

When the GCD is large, listing every divisor can produce thousands of values and stall a tab. This tool therefore enumerates common factors only when the GCD is at most 1,000,000,000. Homework GCDs are almost always far smaller than that. For a prime-power breakdown of a single integer - not just of the GCD - use the prime factorization calculator.

The identity gcd(a, b) × lcm(a, b) = |a · b| holds for two positive integers. It does not extend in the same one-line way to three numbers: gcd(48, 18, 30) × lcm(48, 18, 30) = 6 × 720 = 4320, while 48 × 18 × 30 = 25920. Do not use the two-number product check on a triple. Compute LCM by reducing pairwise instead.

The reason gcd(a, b) = gcd(b, a mod b) is that any common divisor of a and b also divides a − qb for any integer q, and in particular divides the remainder a mod b. Working in the other direction, any common divisor of b and the remainder divides a = qb + remainder. So the common-divisor sets of (a, b) and of (b, a mod b) are identical, and so is the greatest one. Repeating the step shrinks the numbers until a remainder of 0 appears; the previous remainder is then the GCD.

Prime factorizations give the same GCD another way: write each number as a product of primes and keep the minimum exponent for every prime. 48 = 2⁴ × 3, 18 = 2 × 3², 30 = 2 × 3 × 5, so the GCD is 2¹ × 3¹ = 6. The LCM keeps the maximum exponents: 2⁴ × 3² × 5 = 16 × 9 × 5 = 720. Factor trees are excellent for small homework numbers and painful for large ones; Euclid does not care how many prime factors hide inside the inputs.

Coprime integers (GCD 1) still have LCM equal to their product. That is why 8 and 15 give LCM 120, and why two distinct primes always have LCM equal to that product. If every number in a list is a multiple of the same integer k, you can divide each by k, compute GCD or LCM of the smaller list, then scale back: gcd(ka, kb) = k · gcd(a, b) for positive k.

A frequent homework trap is listing factors of only one number. The factors of 48 include 8 and 16, but those do not divide 18, so they are not common. Another trap is treating GCD as “the overlap of the prime lists” without taking minimum exponents: 48 and 18 both contain a 2, but 48 has four factors of 2 while 18 has one, so the GCD keeps only 2¹. A third trap is computing LCM by multiplying all three inputs and dividing by the GCD once - that overcounts. Always fold LCM pairwise: lcm(lcm(48, 18), 30) = lcm(144, 30) = 720.

When two numbers share no prime factors, students sometimes write GCD 0. Zero is a multiple of every integer, but it is not a divisor in the school-book sense used here, and this calculator rejects 0 as an input. Coprime pairs have GCD 1, never 0. Likewise LCM is never smaller than each input: lcm is at least the maximum of the numbers. If a result shows LCM 12 for inputs 12 and 18, that is wrong (the true LCM is 36); use the identity a × b = gcd × lcm on a pair to catch that class of error in a few seconds.