Find the lowest common multiple of two or more numbers — the smallest number that all of them divide evenly into.
The greatest common divisor of the pair is found first, using the Euclidean algorithm.
The product of the two numbers is divided by that GCD.
Dividing before multiplying keeps the intermediate numbers small and avoids overflow on large inputs.
LCM: |a × b| ÷ GCD(a, b)
For a list: Apply the same rule across the numbers in turn
Finding the LCM of 4 and 6.
Numbers: 4, 6
GCD of 4 and 6 is 2
Multiply: 4 × 6 = 24
Divide by the GCD: 24 ÷ 2 = 12
Answer: LCM = 12
An LCM calculator finds the lowest common multiple of two or more whole numbers, the smallest positive number that every one of them divides exactly. It works through the list pair by pair, first finding the HCF of each pair and then using LCM = a × b ÷ HCF, and shows each stage.
The LCM answers 'when do things line up again' questions. Three school bells that ring every 12, 15 and 20 minutes ring together again after the LCM of those numbers. Buses on different routes, traffic signals, shift rotations and medicine schedules follow the same logic. In arithmetic, the LCM of the denominators is the lowest common denominator used to add and compare fractions.
1. Take the first two numbers a and b.
2. Find HCF(a, b), for example by Euclid's division algorithm.
3. Compute LCM(a, b) = (a × b) ÷ HCF(a, b). Dividing one number by the HCF before multiplying keeps the working small.
4. Take this LCM and the next number in the list, and repeat the two steps.
5. Continue until the list is used up; the last result is the LCM of all the numbers.
6. Check that each original number divides the answer exactly.
Multiplying a and b always gives a common multiple, but it counts shared factors twice. For 12 = 2² × 3 and 15 = 3 × 5, the product 180 includes the common 3 twice. Dividing by the HCF, 3, removes the duplicate and leaves 60 = 2² × 3 × 5, which contains every prime to exactly the power needed. That is the smallest common multiple. When a and b are coprime, the HCF is 1 and the LCM is simply the product.
The LCM is associative: LCM(a, b, c) = LCM(LCM(a, b), c). This works because a number is a common multiple of a, b and c exactly when it is a multiple of LCM(a, b) and of c. So the calculator folds the list from left to right, and the order does not change the final answer. In prime factor form, the LCM of a list takes every prime that appears in any number, raised to its highest power in the list.
The LCM is always at least as large as the biggest number, while the HCF is at most as large as the smallest. If a problem asks when events coincide, how long until a repeat, or what is the smallest quantity that can be shared evenly among different group sizes, it wants the LCM. If it asks for the largest equal part, group or measure, it wants the HCF. Checking which way the answer should move relative to the inputs catches most mix-ups.
At Kiran's school, three bells are set to ring every 12, 15 and 20 minutes, and they have just rung together at the start of the day.
HCF of 12 and 15: Needed before applying the LCM formula
LCM of 12 and 15: (12 × 15) ÷ 3
HCF of 60 and 20: Needed before applying the LCM formula
LCM of 60 and 20: (60 × 20) ÷ 20
Answer: LCM (lowest common multiple) 60; HCF (for comparison) 1; Numbers used 12, 15, 20
Multiplying all the numbers together and calling the product the LCM.
Using the LCM when the question asks for the largest equal group, which needs the HCF.
Forgetting to convert the answer back into clock time or dates when the question asks when events coincide.
Taking the lowest power of each prime instead of the highest in the factor method.
Finding the lowest common denominator to add or compare fractions.
Working out when bells, lights or buses with different intervals coincide.
Planning shift or duty rotations that repeat on different cycles.
Finding the smallest number of sweets that can be shared equally among groups of different sizes.
Gear and pulley problems where teeth counts must line up again.
Where is the LCM actually used?
Most often, it is used for adding fractions, where the lowest common denominator is the LCM of the bottom numbers. It also helps with timing questions — two buses leaving every 4 and 6 minutes will leave together again every 12 minutes.
Is the LCM always the two numbers multiplied together?
Only when they are coprime (share no common factor). 7 and 13 share no factors, so their LCM is 91. But 4 and 6 share a factor of 2, so their LCM is 12, not 24.