How to Find the Least Common Multiple (LCM): Step-by-Step Guide
The Least Common Multiple (LCM), also called the Lowest Common Multiple or लघुत्तम समापवर्त्य (ल.स.) in Hindi, is the smallest positive integer that is divisible by all given numbers without leaving a remainder. It is the core mathematical tool used when adding or subtracting fractions with unlike denominators, planning repeating event cycles, and solving modular algebra.
Method 1: Prime Factorization (Powers Method)
The most reliable universal technique for two or more integers:
- Find the prime factor tree of each number.
- Write each number in exponential form (e.g., 12 = 2² × 3¹).
- Identify all distinct prime factors present in any number.
- For each prime, select the highest exponent (power).
- Multiply these prime powers together to get the LCM.
Method 2: Common Continuous Division (Ladder Method)
The standard method taught in school curriculums across India and globally:
- Write the numbers side-by-side separated by commas.
- Divide simultaneously by the smallest prime that divides at least two numbers.
- Bring down any numbers that are not divisible unchanged.
- Continue until the remaining row numbers are mutually coprime.
- Multiply all divisor primes on the left and the final row quotients.
Example: Finding the LCM of 12 and 18
Prime factor breakdown:
12 = 2² × 3¹
18 = 2¹ × 3²
- Highest power of prime 2:
2² = 4 - Highest power of prime 3:
3² = 9 - LCM Calculation:
LCM(12, 18) = 2² × 3² = 4 × 9 = 36 - Formula Verification using HCF:
LCM = (12 × 18) / HCF = 216 / 6 = 36
Frequently Asked Questions (LCM FAQs)
Can LCM ever be smaller than the given numbers?
No. The LCM is a multiple, meaning it must be equal to or greater than the largest integer in your set. For instance, LCM(4, 12) is 12, never less than 12.
What is the formula linking LCM and HCF?
For any two numbers a and b: LCM(a, b) × HCF(a, b) = a × b. Therefore, LCM(a, b) = (a × b) / HCF(a, b). Note: This product rule holds strictly for two numbers; for three or more numbers, calculate pairwise or use prime factorization.
How to calculate LCM of 3 numbers (e.g. 25, 39, 60)?
Write out their prime factors: 25 = 5², 39 = 3 × 13, and 60 = 2² × 3 × 5. Take the highest powers of all primes: 2², 3¹, 5², and 13¹. Multiply them: 4 × 3 × 25 × 13 = 3,900.