LCM & HCF

Understanding and applying LCM and HCF effectively requires strong analytical and reasoning skills, which are highly valued in competitive exams like CAT, GATE, SSC, and Bank exams. Solving problems involving these concepts tests your ability to identify patterns, make logical deductions, and apply mathematical principles efficiently

LCM and HCF are not limited to number theory. They have applications in a variety of topics tested in competitive exams, such as:

* Algebra: Simplifying expressions, solving equations and inequalities, and working with modular arithmetic.

* Geometry: Calculating ratios of areas and volumes, and finding geometric relationships.

* Combinatorics: Counting arrangements and permutations, and solving probability problems

HCF and LCM Definition

HCF (Highest Common Factor):

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. In other words, it is the greatest number that divides all the given numbers.

For example, consider the numbers 12 and 18:

The factors of 12 are 1, 2, 3, 4, 6, and 12.
The factors of 18 are 1, 2, 3, 6, 9, and 18.

The common factors are 1, 2, 3, and 6. Among these, 6 is the largest, so the HCF of 12 and 18 is 6.

LCM (Least Common Multiple):

The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by each of the numbers without leaving a remainder. In other words, it is the smallest common multiple of the given numbers.

For example, consider the numbers 4 and 5:

The multiples of 4 are 4, 8, 12, 16, 20, 24, ...
The multiples of 5 are 5, 10, 15, 20, 25, 30, ...

The common multiples are 20, 40, 60, ... Among these, 20 is the smallest, so the LCM of 4 and 5 is 20

LCM and HCF of Two Numbers

The Least Common Multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers without leaving a remainder. There are different methods to find the LCM, but one common approach involves the prime factorization of each number.

Let's consider two numbers, $a$ and $b$ , and find their LCM:

Prime Factorization Method:

a. Find the prime factorization of each number.

b. Identify all the prime factors involved.

c. The LCM is obtained by multiplying the highest power of each prime factor involved.

Example:

Find the LCM of 12 and 18.

Prime factorization of 12: $2$
Prime factorization of 18:

The LCM is $= 36$

Division Method:

a. Divide the product of the two numbers by their HCF (Highest Common Factor)

Using the List Method:

a. Make a list of multiples of the larger number until you find a multiple that is divisible by the smaller number.

Example:

Find the LCM of 8 and 12.

Multiples of 8: 8, 16, 24, 32, 40, ...
Multiples of 12: 12, 24, 36, 48, ...

The LCM is 24

HCF of Two Numbers

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two numbers is the largest positive integer that divides both numbers without leaving a remainder. There are various methods to find the HCF, but one common approach involves the prime factorization of each number.

Let's consider two numbers, $a$ and $b$ , and find their HCF:

Prime Factorization Method:

a. Find the prime factorization of each number.

b. Identify all the common prime factors involved.

c. The HCF is obtained by multiplying these common prime factors.

Example:

Find the HCF of 36 and 48.
- Prime factorization of 36:
- Prime factorization of 48: $2$
The common prime factors are and , so the HCF is $12$ .
Division Method:

a. Use the division method by repeatedly dividing the larger number by the smaller number until the remainder is zero.

Example:

Find the HCF of 72 and 120.
- Divide 120 by 72: $120 = 72 \times 1 + 48$
- Divide 72 by 48: $72 = 48 \times 1 + 24$
- Divide 48 by 24: $48 = 24 \times 2 + 0$
The last non-zero remainder is 24, so the HCF is 24.
Using the List Method:

a. Make a list of the factors of both numbers and identify the common factors.

Example:

Find the HCF of 15 and 25.
- Factors of 15: 1, 3, 5, 15
- Factors of 25: 1, 5, 25
The common factor is 5, so the HCF is 5

Prime Factorization Method

HCF by Prime Factorization Method

To find the Highest Common Factor (HCF) of two numbers using the Prime Factorization Method, follow these steps:

Step 1: Prime Factorization of Each Number

Write down the prime factorization of each of the given numbers.

Example: Find the HCF of 36 and 48

Prime factorization of 36:
Prime factorization of 48:

Step 2: Identify Common Prime Factors

Identify the common prime factors shared by both numbers.

In this example:

Common factor of 2: (since is the highest power shared by both 36 and 48).
Common factor of 3: (since is the highest power shared by both 36 and 48).

Step 3: Multiply Common Prime Factors

Multiply the identified common prime factors to find the HCF.

$H CF = 2^{2} \times 3 = 12$

Answer:

The HCF of 36 and 48 is $12$

$LCM by Prime Factorization Method$

To find the Least Common Multiple (LCM) of two numbers using the Prime Factorization Method, follow these steps:

Step 1: Prime Factorization of Each Number

Write down the prime factorization of each of the given numbers.

Example: Find the LCM of 12 and 18

Prime factorization of 12:
Prime factorization of 18:

Step 2: Identify All Prime Factors

Identify all the prime factors involved by listing them.

In this example:

Prime factors:

Step 3: Multiply Highest Powers of Prime Factors

Multiply the highest powers of each prime factor involved.

$2^{2} \times 3^{2} = 36$

Answer:

The LCM of 12 and 18 is $36$

Practice Questions on LCM and HCF

Question 1: Find the HCF of 42 and 56.

Solution:

Prime factorization of 42:
Prime factorization of 56:

Common prime factors:

HCF =

Answer: The HCF of 42 and 56 is 14

Question 2: Find the LCM of 15 and 20.

Solution:

Prime factorization of 15:
Prime factorization of 20:

All prime factors:

LCM =

Answer: The LCM of 15 and 20 is 60

Question 3: Find the LCM and HCF of 18 and 24.

Solution: To find the LCM and HCF, we can use the prime factorization method.

Prime factorization of 18: $2 \times 3^{2}$

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

$L CM (18, 24) = 2^{3} \times 3^{2} = 72$

$H CF (18, 24) = 2 \times 3 = 6$

So, the LCM of 18 and 24 is 72, and the HCF is 6.

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LCM & HCF

LCM & HCF

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