LCM & HCF Long Division Method Calculator

LCM & HCF calculator: enter two whole numbers to get HCF (by division) and LCM.

Output: HCF and LCM with Euclidean division steps.

The LCM & HCF Long Division Method Calculator finds the least common multiple and the highest common factor using the division method. For the LCM, it divides the numbers together by common prime factors until only 1s remain, then multiplies all the divisors and leftovers. For the HCF, it uses successive division — dividing the larger number by the smaller, then the divisor by the remainder — until the remainder is 0. Enter two or more numbers to see the division ladder, every prime divisor, and the final LCM and HCF with each step shown.

How to use the LCM & HCF Long Division Method Calculator

To find the LCM or HCF by division, follow these 4 steps:

Enter the numbers you want to work with, separated by spaces or commas.
Choose LCM (common division ladder) or HCF (successive division).
Click Calculate to run the division method.
Read the LCM or HCF along with every divisor and remainder shown.

For the LCM, keep dividing by a prime that divides at least one number; for the HCF, keep replacing the pair with (divisor, remainder) until the remainder is 0.

LCM and HCF by division

The highest common factor (HCF, also called GCD) is the largest number that divides every input exactly, and the least common multiple (LCM) is the smallest number that every input divides into. The division method finds them without listing all factors or multiples. The HCF uses the Euclidean idea that any common factor of two numbers also divides their remainder, so repeated division shrinks the problem to the answer. The LCM ladder strips out shared prime factors once and unique factors as they appear, which is why multiplying the divisors and leftovers rebuilds the smallest common multiple.

Division shares a total into equal groups. Long division does this digit by digit.

Divisor (32) — the number you divide by. Place it to the left of the bracket.

How LCM and HCF work by long division

The calculator finds both results through five internal actions:

For the HCF, divides the larger number by the smaller and records the remainder.
Replaces the pair with (previous divisor, remainder) and repeats until the remainder is 0.
Reports the last non-zero divisor as the HCF.
For the LCM, divides all numbers together by a common prime, carrying down any number the prime does not divide.
Continues until every column is 1, then multiplies all the divisors and any leftovers to get the LCM.

The two methods share the division discipline but read the answer differently: the HCF is the last divisor, and the LCM is the product of all divisors.

For the HCF, divides the larger number by the smaller and records the remainder.

LCM and HCF relationships

The calculator uses the Euclidean rule for the HCF: HCF(a, b) = HCF(b, a mod b), stopping when the remainder is 0. It uses the product rule linking both: LCM(a, b) × HCF(a, b) = a × b. For 36 and 48, HCF = 12 and LCM = 144, and the check is 144 × 12 = 36 × 48 = 1728.

Dividend

Divisor

Quotient

Remainder

48 = 18 × 2 + 12 → Identity holds

LCM & HCF example problems

These examples show successive division for HCF and the ladder for LCM.

Example 1 — HCF by division: 48 and 36

Divide 48 by 36: quotient 1, remainder 12.
Divide 36 by 12: quotient 3, remainder 0.
The last non-zero divisor is 12, so HCF(48, 36) = 12.

Example 2 — LCM by division ladder: 12 and 18

Divide by 2: 12 → 6, 18 → 9. Divide by 3: 6 → 2, 9 → 3.
Divide by 2: 2 → 1, 3 stays 3. Divide by 3: 3 → 1.
Multiply the divisors 2 × 3 × 2 × 3 = 36, so LCM(12, 18) = 36.

Example 3 — Using the product rule: 15 and 20

HCF by division: 20 ÷ 15 = 1 R5, then 15 ÷ 5 = 3 R0, so HCF = 5.
Apply LCM × HCF = 15 × 20 = 300, so LCM = 300 ÷ 5 = 60.
So HCF(15, 20) = 5 and LCM(15, 20) = 60.

Divide 48 by 36: quotient 1, remainder 12.
Divide 36 by 12: quotient 3, remainder 0.
The last non-zero divisor is 12, so HCF(48, 36) = 12.

Worked LCM & HCF problems

How do you find the HCF of 84 and 120 by division?

HCF(84, 120) = 12. Divide 120 by 84: remainder 36. Divide 84 by 36: remainder 12. Divide 36 by 12: remainder 0. The last non-zero divisor, 12, is the HCF. Each step uses the fact that a common factor of two numbers also divides their remainder.

4818

How do you find the LCM of 8, 12, and 15?

LCM(8, 12, 15) = 120. Divide by 2: 8 → 4, 12 → 6, 15 stays. Divide by 2: 4 → 2, 6 → 3, 15 stays. Divide by 2: 2 → 1, 3 and 15 stay. Divide by 3: 3 → 1, 15 → 5. Divide by 5: 5 → 1. Multiply 2 × 2 × 2 × 3 × 5 = 120.

3624

Common LCM & HCF mistakes

The division method produces 5 frequent errors:

Reporting the last remainder instead of the last non-zero divisor as the HCF.
Multiplying only the shared divisors and forgetting leftover factors for the LCM.
Dividing by a number that is not prime, which double-counts factors.
Stopping the LCM ladder before every column reaches 1.
Swapping the results — giving the LCM when the HCF was asked, or vice versa.

The LCM & HCF Long Division Method Calculator tracks each divisor and remainder, so it reports the correct answer for whichever you request.

Reporting the last remainder instead of the last non-zero divisor as the HCF.

Multiplying only the shared divisors and forgetting leftover factors for the LCM.

Dividing by a number that is not prime, which double-counts factors.

Stopping the LCM ladder before every column reaches 1.

Swapping the results — giving the LCM when the HCF was asked, or vice versa.

Frequently Asked Questions

How do you find the HCF using the long division method?

Divide the larger number by the smaller and note the remainder, then divide the previous divisor by that remainder, repeating until the remainder is 0. The last non-zero divisor is the HCF.

How do you find the LCM using the division method?

Write the numbers in a row and divide them together by a common prime, carrying down any number the prime does not divide. Continue until all are 1, then multiply every divisor to get the LCM.

What is the difference between LCM and HCF?

The HCF is the largest number that divides all the inputs, while the LCM is the smallest number that all inputs divide into. For 12 and 18, HCF = 6 and LCM = 36.

How are LCM and HCF related?

For two numbers, LCM × HCF = the product of the numbers. So once you have one, you can find the other: LCM = (a × b) ÷ HCF.

Why is the last non-zero divisor the HCF?

Because any common factor of two numbers also divides their remainder, so the common factors are preserved at every division step. When the remainder hits 0, the current divisor divides both numbers exactly.

Can I find the HCF of more than two numbers?

Yes. Find the HCF of the first two, then take the HCF of that result with the next number, and so on. The final result is the HCF of all of them.

Do I have to divide by prime numbers for the LCM?

Yes, dividing by primes in the ladder avoids double-counting factors and gives the correct LCM. Using a composite divisor can skip factors that are still needed.

The LCM–HCF relationship

The LCM & HCF Long Division Method Calculator uses the Euclidean rule HCF(a, b) = HCF(b, a mod b) and the product rule LCM(a, b) × HCF(a, b) = a × b. For 48 and 36, HCF = 12 and LCM = 144, and the identity confirms 144 × 12 = 48 × 36 = 1728.