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LCM

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Introduction to LCM

Before we dive into the concept of LCM, let's start with something familiar: multiples. A multiple of a number is what you get when you multiply that number by any natural number. For example, multiples of 3 are 3, 6, 9, 12, and so on.

Sometimes, when working with two or more numbers, we need to find a number that is a multiple of all those numbers. Such numbers are called common multiples. Among these common multiples, the smallest positive one is called the Least Common Multiple (LCM).

Understanding LCM is important because it helps solve problems involving synchronization, scheduling, measurements, and currency denominations, which are common in competitive exams.

Definition and Basic Understanding of LCM

The Least Common Multiple (LCM) of two or more numbers is the smallest number (other than zero) that is exactly divisible by all of them.

Let's understand this with an example:

Multiples of 4 and 6
Multiples of 4 Multiples of 6
46
812
1218
1624
2030
2436

Looking at the table, the common multiples of 4 and 6 are 12, 24, 36, ... The smallest among these is 12. Therefore, LCM(4, 6) = 12.

Key Concept

Least Common Multiple (LCM)

The smallest positive number divisible by all given numbers.

Methods to Find LCM

There are three main methods to find the LCM of numbers. Each method has its own advantages depending on the numbers involved.

graph TD    A[Start] --> B{Choose Method}    B --> C[Listing Multiples Method]    B --> D[Prime Factorization Method]    B --> E[Division Method]    C --> C1[List multiples of each number]    C1 --> C2[Identify common multiples]    C2 --> C3[Select smallest common multiple as LCM]    D --> D1[Find prime factors of each number]    D1 --> D2[Take highest powers of all primes]    D2 --> D3[Multiply these to get LCM]    E --> E1[Write numbers in a row]    E1 --> E2[Divide by common prime factors]    E2 --> E3[Repeat until all numbers become 1]    E3 --> E4[Multiply all divisors to get LCM]

Relation between LCM and GCD

LCM is closely related to another important concept called the Greatest Common Divisor (GCD), which is the largest number that divides two numbers exactly.

The relationship between LCM and GCD of two numbers \(a\) and \(b\) is given by the formula:

LCM and GCD Relationship

\[\text{LCM}(a,b) \times \text{GCD}(a,b) = |a \times b|\]

The product of LCM and GCD of two numbers equals the product of the numbers themselves.

a,b = Two integers
\(\text{LCM}(a,b)\) = Least Common Multiple
\(\text{GCD}(a,b)\) = Greatest Common Divisor

This formula is very useful when you know the GCD and want to find the LCM quickly, or vice versa.

Number a Number b GCD LCM Product of a and b Divides

Worked Examples

Example 1: LCM by Listing Multiples Easy
Find the LCM of 12 and 15 by listing their multiples.

Step 1: List the first few multiples of 12:

12, 24, 36, 48, 60, 72, ...

Step 2: List the first few multiples of 15:

15, 30, 45, 60, 75, 90, ...

Step 3: Identify the common multiples:

60, 120, ...

Step 4: The smallest common multiple is 60.

Answer: LCM(12, 15) = 60

Example 2: LCM by Prime Factorization Medium
Find the LCM of 18 and 24 using prime factorization.

Step 1: Find prime factors of 18:

18 = 2 x 3 x 3 = \(2^1 \times 3^2\)

Step 2: Find prime factors of 24:

24 = 2 x 2 x 2 x 3 = \(2^3 \times 3^1\)

Step 3: Take highest powers of each prime:

  • For 2: highest power is \(2^3\)
  • For 3: highest power is \(3^2\)

Step 4: Multiply these highest powers:

LCM = \(2^3 \times 3^2 = 8 \times 9 = 72\)

Answer: LCM(18, 24) = 72

Example 3: Synchronizing Bus Timings Medium
Two buses arrive at a bus stop every 20 minutes and 30 minutes respectively. If both arrive together at 9:00 AM, when will they next arrive together?

Step 1: Find the LCM of 20 and 30 to determine the interval when both arrive together.

Prime factorization:

  • 20 = \(2^2 \times 5\)
  • 30 = \(2 \times 3 \times 5\)

Take highest powers:

  • 2: \(2^2\)
  • 3: \(3^1\)
  • 5: \(5^1\)

LCM = \(2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60\) minutes

Step 2: Add 60 minutes to 9:00 AM.

Next common arrival time = 10:00 AM

Answer: Both buses will next arrive together at 10:00 AM.

Example 4: LCM of Three Numbers by Division Method Hard
Find the LCM of 8, 12, and 20 using the division method.

Step 1: Write the numbers in a row:

8     12     20

Step 2: Divide by the smallest prime that divides at least one number:

Divide by 2:

4     6     10

Divisor so far: 2

Step 3: Divide again by 2:

2     3     5

Divisor so far: 2 x 2 = 4

Step 4: Divide by 2 again (only divides 2 and 5? No, 5 is not divisible by 2), so divide by 3:

2     1     5

Divisor so far: 4 x 3 = 12

Step 5: Divide by 2:

1     1     5

Divisor so far: 12 x 2 = 24

Step 6: Divide by 5:

1     1     1

Divisor so far: 24 x 5 = 120

Step 7: All numbers reduced to 1, stop.

Answer: LCM(8, 12, 20) = 120

Example 5: Real-life Problem Involving Currency (INR) Medium
A shopkeeper wants to pay a certain amount equally using only 50 INR and 75 INR notes. What is the minimum amount he can pay?

Step 1: Find the LCM of 50 and 75 to determine the smallest amount divisible by both denominations.

Prime factorization:

  • 50 = \(2 \times 5^2\)
  • 75 = \(3 \times 5^2\)

Take highest powers:

  • 2: \(2^1\)
  • 3: \(3^1\)
  • 5: \(5^2\)

LCM = \(2 \times 3 \times 25 = 150\)

Answer: The minimum amount payable equally using 50 INR and 75 INR notes is 150 INR.

Formula Bank

LCM of Two Numbers
\[ \text{LCM}(a,b) = \frac{|a \times b|}{\text{GCD}(a,b)} \]
where: \(a, b\) = two integers; \(\text{GCD}(a,b)\) = greatest common divisor of \(a\) and \(b\)
LCM of More Than Two Numbers
\[ \text{LCM}(a,b,c) = \text{LCM}(\text{LCM}(a,b), c) \]
where: \(a, b, c\) = integers

Tips & Tricks

Tip: Use prime factorization for large numbers to avoid listing long multiples.

When to use: When numbers are large or listing multiples is time-consuming.

Tip: Remember the formula \(\text{LCM} \times \text{GCD} = \text{product of the two numbers}\) to quickly find LCM if GCD is known.

When to use: When GCD is easier to calculate or already known.

Tip: For three or more numbers, find LCM iteratively using pairwise LCM calculations.

When to use: When dealing with multiple numbers.

Tip: Check divisibility rules first to simplify numbers before factorization.

When to use: To speed up prime factorization.

Tip: Use the division method for finding LCM of multiple numbers efficiently.

When to use: When dealing with three or more numbers.

Common Mistakes to Avoid

❌ Confusing LCM with GCD and calculating the smaller common divisor instead of the smallest common multiple.
✓ Understand that LCM is the smallest number divisible by both numbers, whereas GCD is the largest number dividing both.
Why: Students often mix up the two concepts due to similar terminology.
❌ Listing multiples incompletely or missing common multiples.
✓ List enough multiples to ensure the smallest common multiple is found.
Why: Students stop listing too early or overlook some multiples.
❌ Incorrect prime factorization leading to wrong LCM.
✓ Double-check prime factors and their highest powers before calculating LCM.
Why: Errors in factorization propagate to wrong answers.
❌ Not applying the LCM x GCD = product formula correctly, especially with negative numbers.
✓ Use absolute values of numbers when applying the formula.
Why: Negative signs confuse the calculation.
❌ Forgetting to include all prime factors at their highest power when combining for LCM.
✓ Take the highest power of each prime factor appearing in any number.
Why: Students sometimes take lower powers, resulting in smaller than actual LCM.
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