Lowest Common Multiple (LCM) | Definition, Formula & Solved Examples

The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive integer divisible by each number. It is commonly used to solve problems involving fractions, ratios, and synchronizing repeating events. LCM is an essential mathematical concept and useful for finding common denominators and aligning cycles.

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1. Introduction to the Lowest Common Multiple:

The Lowest Common Multiple (LCM) is a fundamental arithmetic and number theory concept. It is often used to simplify problems involving fractions, ratios, and various real-life situations like scheduling or synchronization of repeating events. Understanding how to calculate the LCM helps solve a wide range of problems in mathematics, particularly those involving multiples and factors.

2. What is the Lowest Common Multiple:

The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by each of the given numbers. In other words, it is the smallest multiple that two or more numbers share. The LCM is useful in solving problems that require aligning intervals, such as when two events happen at different intervals, and you need to know when they will occur together.

For example, the LCM of $6$ and $8$ is $24$ because $24$ is the smallest number divisible by $6$ and $8$.

3. How to Find the Lowest Common Multiple:

Several methods exist for finding the Lowest Common Multiple (LCM) of two or more numbers. Below are the three primary methods: Listing Method, Prime Factorization Method, and Division Method. Each method works best depending on the size of the numbers and the context in which you're working.

1. Finding LCM Using Listing Method

The listing method is the simplest way to find the LCM, especially for smaller numbers. It involves listing the multiples of the numbers until a common multiple is found.

Steps:

1. List the multiples of each number.

2. Identify the smallest multiple that appears in all lists.

Example: Find the LCM of $4$ and $6$:

• Multiples of $4$: $4, 8, 12, 16, 20, 24, 28, \dots$

• Multiples of $6$: $6, 12, 18, 24, 30, 36, \dots$

The smallest common multiple is $12$, so the LCM of $4$ and $6$ is $12$.

2. Finding LCM Using Prime Factorization Method

The prime factorization method involves breaking down each number into its prime factors and then combining the highest powers of all prime factors involved.

Steps:

1. Find the prime factorization of each number.

2. Identify the highest powers of each prime factor that appear in any of the numbers.

3. Multiply these prime factors together to get the LCM.

Example: Find the LCM of $12$ and $18$.

1. Prime factorization of $12$: $12 = 2^2 \times 3$

2. Prime factorization of $18$: $18 = 2 \times 3^2$

Now, take the highest powers of each prime factor:

• For $2$: $2^2$

• For $3$: $3^2$

LCM $= 2^2 \times 3^2 = 4 \times 9 = 36$.

Thus, the LCM of $12$ and $18$ is $36$.

3. Finding LCM Using Division Method

The division method is another efficient way to find the LCM, especially for larger numbers. It involves dividing the numbers by their common prime factors until no common factors remain, then multiplying the divisors.

Steps:

1. Write the numbers side by side.

2. Divide the numbers by the smallest common prime factor.

3. Continue dividing until no common factors remain.

4. Multiply all divisors to get the LCM.

Example: Find the LCM of $16$ and $24$.

1. Divide by $2$: $\dfrac{16}{2} = 8$, $\dfrac{24}{2} = 12$

2. Divide by $2$ again: $\dfrac{8}{2} = 4$, $\dfrac{12}{2} = 6$

3. Divide by $2$ again: $\dfrac{4}{2} = 2$, $\dfrac{6}{2} = 3$

4. Divide by $2$ again: $\dfrac{2}{2} = 1$, $\dfrac{3}{2}$ ($3$ is no longer divisible by $2$)

Now, multiply all divisors: $2 \times 2 \times 2 \times 2 \times 3 = 48$.

Thus, the LCM of $16$ and $24$ is $48$.

4. Rules for Lowest Common Multiple:

• Rule 1: The LCM of two numbers is always greater than or equal to the larger of the two numbers.

• Rule 2: If one number is a multiple of the other, the LCM is the larger number.

• Rule 3: The LCM of prime numbers is simply the product of the numbers since they have no common factors.

5. Lowest Common Multiple Formula:

The LCM formula for two numbers $a$ and $b$ is derived from the relationship between the LCM and the highest common factor (HCF or GCD):

$\text{LCM}(a, b) = \dfrac{a \times b}{\text{GCD}(a, b)}$

This formula shows that the LCM is the product of the two numbers divided by their HCF.

6. Properties of Lowest Common Multiple:

• Multiplicative Property: The LCM of two numbers is always divisible by both.

• Commutative Property: The LCM of $a$ and $b$ is the same as the LCM of $b$ and $a$.

• Associative Property: The LCM of three or more numbers can be found step by step:
  $\text{LCM}(a, b, c) = \text{LCM}\bigg(\text{LCM}(a, b), c\bigg).$

7. Relationship Between LCM and HCF:

A simple yet powerful formula connects the LCM and HCF of two or more numbers. This formula helps bridge the two concepts and provides a clear relationship between them:

$\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b$

This formula shows that the product of the LCM and HCF of two numbers equals the product of the numbers themselves. This relationship is valuable in understanding the divisibility of numbers and solving various mathematical problems.

Explanation of the Relationship:

• HCF (or GCD): The HCF of two numbers is the largest number that divides both numbers without leaving a remainder. It focuses on the greatest common factor between the numbers.

• LCM: The LCM is the smallest multiple divisible by both numbers. It focuses on finding the smallest number that both numbers divide into evenly.

The relationship can be seen as balancing the divisibility (HCF) and the multiples (LCM) of numbers. When you multiply the LCM by the HCF, the result is the product of the original two numbers.

Example:

Consider the numbers $12$ and $18$.

• Step 1: Find the HCF of $12$ and $18$.

  • Prime factorization of $12$: $2^2 \times 3$

  • Prime factorization of $18$: $2 \times 3^2$

  • The common factor is $2 \times 3 = 6$, so the HCF is $6$.

• Step 2: Find the LCM of $12$ and $18$.

  • Take the highest powers of all primes involved: $2^2 \times 3^2 = 36$, so the LCM is $36$.

• Step 3: Verify the relationship.
  $\text{LCM}(12, 18) \times \text{HCF}(12, 18) = 36 \times 6 = 216$
  $12 \times 18 = 216$

8. Difference Between LCM and HCF:

• LCM (Lowest Common Multiple): The smallest multiple two or more numbers have in common.

• HCF (Highest Common Factor): The largest number that divides two or more numbers without a remainder.

LCM is used when working with multiples and synchronizing cycles, while HCF finds common factors and simplifies fractions.

Example:

Let’s consider the numbers $12$ and $16$.

Step 1: Factors of $12$ and $16$

• Factors of $12$: $1, 2, 3, 4, 6, 12$

• Factors of $16$: $1, 2, 4, 8, 16$

• The highest common factor is $4$.

Step 2: Multiples of $12$ and $16$

• Multiples of $12$: $12, 24, 36, 48, 60, 72, 84, 96, \dots$

• Multiples of $16$: $16, 32, 48, 64, 80, 96, \dots$

• The lowest common multiple is $48$.

Thus:

• HCF of $12$ and $16 = 4$
• LCM of $12$ and $16 = 48$

9. Lowest Common Multiple Solved Examples:

Question: 1

Find the LCM of $15$, $20$ and $26$ using prime factorization.

Solution:

Prime factorization:

• $15 = 3 \times 5$
• $20 = 2^2 \times 5$
• $26 = 2 \times 13$

LCM $= 2^2 \times 3 \times 5 \times 13 = 780$

Question: 2

Find the LCM of $9$ and $12$ using the listing method.

Solution:

• Multipliers of $9 =9,18,27,36,45,54, \dots$
• Multipliers of $12 =12,24,36,48, \dots$

The smallest common multiple is $36$.

Question: 3

Find the LCM of $18$ and $24$. using the division method.

Solution:

1. Divide by $2$: $\dfrac{18}{2} = 9$, $\dfrac{24}{2} = 12$

2. Divide by $2$ again: $9$, $\dfrac{12}{2} = 6$ ($9$ is no longer divisible by 2$)

3. Divide by $2$ again: $9$, $\dfrac{6}{2} = 3$ ($9$ is no longer divisible by 2$)

4. Divide by $3$: $\dfrac{9}{3} = 3$, $\dfrac{3}{3}=1$

5. Divide by $3$ again: $\dfrac{3}{3} = 1$

Now, multiply all divisors: $2 \times 2 \times 2 \times 3 \times 3 = 72$.

Thus, the LCM of $18$ and $24$ is $72$.

Question: 4

Find the LCM of $24$ and $36$ using the formula.

Step-by-Step Solution:

Step 1: Find the HCF of $24$ and $36$
To find the HCF, we will use the prime factorization method.

• Prime factorization of $24$:
  $24 = 2^3 \times 3$

• Prime factorization of $36$:
  $36 = 2^2 \times 3^2$

Now, take the lowest powers of all common prime factors:

• The lowest power of $2$ is $2^2$.
• The lowest power of $3$ is $3^1$.

Therefore, the HCF is:
$HCF(24, 36) = 2^2 \times 3^1 = 4 \times 3 = 12$

Step 2: Apply the LCM formula
Now, use the formula for LCM:

$LCM(24, 36) = \dfrac{24 \times 36}{HCF(24, 36)}$

Substitute the values:

$LCM(24, 36) = \dfrac{24 \times 36}{12}$

Step 3: Simplify the expression

First, multiply $24$ and $36$:

$24 \times 36 = 864$

Now, divide $864$ by $12$:

$\dfrac{864}{12} = 72$

Final Answer:

The LCM of $24$ and $36$ is $72$.

$LCM(24, 36) = 72$

10. Practice Questions on Lowest Common Multiple:

Q:1. Find the LCM of $14$ and $35$ using the prime factorization method.

Q:2. What is the LCM of $6$, $8$, and $12$ using the division method?

Q:3. Calculate the LCM of $21$ and $28$ using the listing method.

Q:4. Find the LCM of $9$, $18$, and $27$ using the lCM formula.

11. FAQs on Lowest Common Multiple:

What is the Lowest Common Multiple (LCM)?

The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive integer divisible by each given number. It’s useful for solving problems involving fractions, ratios, and scheduling events.

How do you find the LCM of two numbers?

The LCM can be found using various methods such as the listing method, prime factorization, or the division method. You can also use the LCM formula: $LCM(a, b) = \dfrac{a \times b}{HCF(a, b)}$, where HCF is the highest common factor.

Can the LCM of two numbers be smaller than one of the numbers?

No, the LCM of two numbers is always greater than or equal to the largest number. The LCM is the smallest multiple common to both numbers.

What is the difference between LCM and HCF?

The LCM is the smallest common multiple of two or more numbers, while the HCF (Highest Common Factor) is the largest number that divides two or more numbers without leaving a remainder. LCM deals with multiples, while HCF deals with factors.

What is the LCM of two prime numbers?

The LCM of two prime numbers is always the product of the two numbers since they have no common factors other than 1. For example, the LCM of $5$ and $7$ is $5 \times 7 = 35$.

Is the LCM of any number and 1 always the number itself?

Yes, the LCM of any number, and $1$ is always the number itself because $1$ is a divisor of all numbers. For example, $LCM(1,7)=7.$

How is LCM used in real life?

LCM is used in real-life situations, such as scheduling events at different intervals (e.g., buses or lights), simplifying fractions, and solving problems that involve aligning repeated patterns or cycles.

Can the LCM of negative numbers be calculated?

LCM is typically calculated for positive integers, but it can also be found for negative numbers by ignoring the negative sign. For example, the LCM of $-4$ and $-6$ is the same as the LCM of $4$ and $6$, which is $12$.

12. Real-life Application of Lowest Common Multiple:

In real life, LCM synchronizes repeating events, such as scheduling. For example, if two buses arrive at intervals of $15$ minutes and $20$ minutes, the LCM of $15$ and $20$ ($60$) will tell you that both buses will arrive together every $60$ minutes. LCM is also used to simplify fractions, especially when adding or subtracting fractions with different denominators.

13. Conclusion:

The Lowest Common Multiple is a crucial mathematical tool for solving many problems, from simplifying fractions to synchronizing schedules. Understanding how to find the LCM using prime factorization, listing, and division allows efficient problem-solving. With its many properties and connections to other mathematical concepts like HCF, LCM is a valuable concept that finds applications in theoretical mathematics and everyday life.

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