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Quantitative AptitudeBeginner4 min study

How to Find the LCM of Three Numbers: Step-by-Step Guide

तीन संख्याओं का ल.स. (LCM) कैसे निकालें: आसान तरीका

The Least Common Multiple (LCM), or लघुत्तम समापवर्त्य (ल.स.), of three numbers is the smallest positive number that is completely divisible by all three numbers without leaving any remainder. In competitive exams like MP Police Constable, MP Patwari, and SSC, LCM questions frequently appear in arithmetic, time and work, and bell-ringing interval problems.

Key Rules & Core Formulas

Understand these foundational properties before solving questions

1Definition of LCM

LCM(a, b, c) = Smallest number divisible by a, b, and c

Every multiple of the LCM is also a common multiple of the three numbers.

The LCM of three positive integers is always greater than or equal to the largest of the three numbers.

2Prime Factorization Rule

LCM = Product of highest powers of all prime factors present

Break down each number into prime factors, then select the highest exponent for every distinct prime factor and multiply them together.

Best method when numbers are relatively small or already given in exponential form.

3Common Division Method (L-Ladder)

Continuous division by common prime factors

Write all three numbers in a row and divide by the smallest prime that divides at least two numbers. Carry down numbers that cannot be divided.

Fastest and most reliable method for competitive exam calculation speed.
Standard Method

Step-by-Step Method to Solve

Follow these universal steps in order to solve any Quantitative Aptitude question accurately:

  1. 1

    Step 1: Write the three numbers in a row

    Place the three given numbers side by side separated by commas inside a division ladder (L-shape table).

    Pro-Tip: Sorting the numbers in ascending order makes it easier to track your division.
  2. 2

    Step 2: Identify the smallest prime factor dividing at least two numbers

    Start with the smallest prime number (2, 3, 5, 7, etc.). Check if it divides at least two of the three numbers completely.

    Pro-Tip: Always test 2 first if any of the numbers are even.
  3. 3

    Step 3: Divide and carry down undivided numbers

    Write the quotients directly below the divisible numbers. If a number is NOT divisible by that prime factor, simply carry it down unchanged to the next row.

    Pro-Tip: A common mistake is forgetting to carry down undivided numbers. Always copy them down!
  4. 4

    Step 4: Repeat until no two numbers share a common prime divisor

    Continue dividing by prime numbers until the bottom row contains numbers that are pairwise co-prime (no two numbers share any common factor other than 1).

    Pro-Tip: You can stop as soon as all remaining bottom numbers are mutually co-prime.
  5. 5

    Step 5: Multiply all divisors and bottom row numbers

    Multiply all the prime numbers on the left-hand column and all the remaining numbers at the bottom row. The result is the LCM.

    Pro-Tip: Double-check your final multiplication by ensuring the answer ends with the expected digit.

Worked Examples & Walkthroughs

Real questions solved step-by-step showing every calculation

1Example 1Method: Common Division Method

Question

Find the LCM of 12, 18, and 30 using the Common Division method.

Step-by-Step Solution:

Step 1: Set up the ladder:Write down: 12, 18, 30
Step 2: Divide by prime 2:All three numbers are even: 12 ÷ 2 = 6, 18 ÷ 2 = 9, 30 ÷ 2 = 15. The new row is: 6, 9, 15.
Step 3: Divide by prime 3:6, 9, and 15 are all divisible by 3: 6 ÷ 3 = 2, 9 ÷ 3 = 3, 15 ÷ 3 = 5. The new row is: 2, 3, 5.
Step 4: Check bottom row:The remaining numbers are 2, 3, and 5. Since these are all prime and share no common factors, we stop dividing.
Step 5: Multiply all factors:LCM = (Divisors on left) × (Bottom row) = (2 × 3) × (2 × 3 × 5) = 6 × 30 = 180.
Final Result:
LCM(12, 18, 30) = 180
Exam Shortcut / Verification: Verification: 180 ÷ 12 = 15, 180 ÷ 18 = 10, 180 ÷ 30 = 6. All divide evenly without remainder.
2Example 2Method: Prime Factorization Method

Question

Find the LCM of 15, 20, and 25 using the Prime Factorization method.

Step-by-Step Solution:

Step 1: Factorize each number:15 = 3¹ × 5¹, 20 = 2² × 5¹, 25 = 5².
Step 2: List all unique prime factors:The distinct prime factors involved are 2, 3, and 5.
Step 3: Select highest powers:Highest power of 2 is 2² = 4. Highest power of 3 is 3¹ = 3. Highest power of 5 is 5² = 25.
Step 4: Multiply highest powers:LCM = 2² × 3¹ × 5² = 4 × 3 × 25 = 300.
Final Result:
LCM(15, 20, 25) = 300
Exam Shortcut / Verification: In exam questions, whenever numbers end in 0 and 5, always expect 5² = 25 in the LCM.

Common Mistakes to Avoid

Avoid these frequent traps to protect yourself from negative marking

1

Stopping the division when only one number is divisible

Why it happens: Students sometimes think they can only divide if ALL three numbers are divisible (confusing LCM with HCF).
Correct approach: For LCM, divide as long as at least TWO numbers share a prime factor. Only in HCF must all numbers be divisible.
2

Forgetting to carry down non-divisible numbers

Why it happens: Rushing through calculation leads to accidentally dropping numbers that were not divisible.
Correct approach: Always write each row with exactly three numbers. If a number cannot be divided, copy it straight down unchanged.
3

Dividing by composite numbers (like 4, 6, or 9)

Why it happens: Trying to skip steps by dividing by larger numbers.
Correct approach: Always divide strictly by prime numbers (2, 3, 5, 7, 11...). This ensures mathematical accuracy every time.
Self-Assessment

Try It Yourself (Practice Questions)

Test your understanding on your own first, then reveal the step-by-step solution to check your work:

Practice #1

Find the LCM of 16, 24, and 36.

Practice #2

Three temple bells toll together at intervals of 9, 12, and 15 minutes respectively. If they toll together now, after how many minutes will they toll together next?

Quick Revision Summary

  • LCM is the smallest number divisible by all three given numbers.
  • Use the Common Division method: divide if at least 2 numbers share a prime factor.
  • Always carry down undivided numbers unchanged.
  • Multiply all divisors on the left and the final row at the bottom to get the answer.
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