How to Solve LCM Problems: Complete Guide

How to Solve LCM Problems: Complete Guide

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Imagine you’re planning a party where two friends visit every 6 days and another every 8 days. When will they BOTH show up on the same day? That’s an LCM problem—and mastering it unlocks real-life scheduling, exam questions on fractions, and even algebra!

What You Need To Know First

Before tackling LCM, you must understand: 1. Prime factorization – Breaking numbers into products of primes (e.g., 12 = 2² × 3). 2. Multiples – The result of multiplying a number by integers (e.g., multiples of 4: 4, 8, 12, 16…). 3. Greatest Common Divisor (GCD) – The largest number that divides two numbers (e.g., GCD of 8 and 12 is 4).

Key Vocabulary

Formulas To Know

1. Prime Factorization Method (MEMORISE THIS)

Formula: For two numbers, A and B: 1. Find the prime factors of A and B. 2. Take the highest power of each prime that appears in either factorization. 3. Multiply these together to get the LCM.

Example: Find LCM of 12 and 18. - 12 = 2² × 3¹ - 18 = 2¹ × 3² - LCM = 2² × 3² = 4 × 9 = 36

2. Division Method (MEMORISE THIS)

Steps: 1. Write the numbers side by side. 2. Divide by the smallest prime that divides at least one number. 3. Write the quotients below. If a number isn’t divisible, bring it down. 4. Repeat until all quotients are 1. 5. Multiply all the divisors to get the LCM.

Example: Find LCM of 6 and 8.

2 | 6, 8
2 | 3, 4
2 | 3, 2
3 | 1, 1

LCM = 2 × 2 × 2 × 3 = 24

3. Using GCD (Given on exam sheet)

Formula: LCM(A, B) = (A × B) ÷ GCD(A, B)

Example: Find LCM of 15 and 20. - GCD(15, 20) = 5 - LCM = (15 × 20) ÷ 5 = 300 ÷ 5 = 60

Step-by-Step Method

Method 1: Prime Factorization (Best for small numbers)

Steps: 1. Write the prime factors of each number. 2. Identify the highest power of each prime that appears. 3. Multiply these together to get the LCM.

Worked Example: Find LCM of 12 and 20. 1. Prime factors:
- 12 = 2² × 3¹
- 20 = 2² × 5¹ 2. Highest powers:
- 2², 3¹, 5¹ 3. LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60

Method 2: Division Method (Best for larger numbers)

Steps: 1. Write the numbers in a row. 2. Divide by the smallest prime that divides at least one number. 3. Write the quotients below. If a number isn’t divisible, bring it down. 4. Repeat until all quotients are 1. 5. Multiply all divisors to get the LCM.

Worked Example: Find LCM of 12, 15, and 20.

2 | 12, 15, 20
2 | 6, 15, 10
3 | 3, 15, 5
5 | 1, 5, 5
  | 1, 1, 1

LCM = 2 × 2 × 3 × 5 = 60

Method 3: Using GCD (Fastest for two numbers)

Steps: 1. Find the GCD of the two numbers. 2. Multiply the numbers together. 3. Divide by the GCD to get the LCM.

Worked Example: Find LCM of 18 and 24. 1. GCD(18, 24) = 6 2. 18 × 24 = 432 3. LCM = 432 ÷ 6 = 72

Worked Examples

Example 1 – Basic (No Tricks)

Problem: Find the LCM of 8 and 12.

Solution: 1. Prime factors:
- 8 = 2³
- 12 = 2² × 3¹ 2. Highest powers: 2³, 3¹ 3. LCM = 2³ × 3 = 8 × 3 = 24

What we did and why: We broke both numbers into primes, took the highest power of each, and multiplied them. This ensures the LCM is divisible by both numbers.

Example 2 – Medium (One Added Complication)

Problem: Find the LCM of 9, 12, and 15.

Solution: 1. Prime factors:
- 9 = 3²
- 12 = 2² × 3¹
- 15 = 3¹ × 5¹ 2. Highest powers: 2², 3², 5¹ 3. LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180

What we did and why: We included all primes from all three numbers, ensuring the LCM is divisible by 9, 12, and 15.

Example 3 – Exam Style (Disguised, Time-Pressure)

Problem: Two buses leave a station at the same time. Bus A returns every 18 minutes, and Bus B returns every 24 minutes. When will they both return at the same time again?

Solution: 1. This is an LCM problem (find the smallest time both buses align). 2. Prime factors:
- 18 = 2¹ × 3²
- 24 = 2³ × 3¹ 3. Highest powers: 2³, 3² 4. LCM = 2³ × 3² = 8 × 9 = 72 minutes

What we did and why: We recognized the real-world scenario as an LCM problem, then applied prime factorization to find the answer.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in—tonight, before your exam, here’s what you need to remember about LCM: 1. Prime factorization method: Break numbers into primes, take the highest power of each, multiply. 2. Division method: Divide by primes until all quotients are 1, then multiply the divisors. 3. GCD shortcut: If you know GCD, LCM = (A × B) ÷ GCD. 4. Watch for traps: Word problems about ‘same time’ or ‘repeating events’ are LCM in disguise. 5. Double-check: Did you take the highest power of each prime? Did you multiply correctly?

Now, go practice 3 problems—one with two numbers, one with three, and one word problem. You’ve got this!