HCF & LCM Concepts

Understanding the Concepts of HCF & LCM

You can now understand the fundamental concepts of HCF and LCM. Listed below are the different HCF and LCM concepts:

1. What is HCF?
2. How to Find the HCF of the given numbers?
3. What is LCM?
4. How to Find the LCM of the given numbers?

1. Highest Common Factor (HCF)

The Highest Common Factor (HCF) of two or more given numbers is the largest number which divides each given number without leaving any remainder.

For example, the HCF of 24 and 36 is 12 because 12 is the largest number which can divide the numbers altogether.

2. How to Find the HCF of the given numbers?

There are multiple methods to find the HCF of the given numbers. Prime factorisation method, L-Division method, and Normal Division method.

2.1 Prime Factorisation method

To find the HCF of the given numbers by prime factorisation,

• We find the prime factors of those numbers.

• After finding the factors, we see the product of the prime factors common to each given number.

Example Problem

Let us find the HCF of 45 and 60 by the prime factorisation method.

The prime factors of 45 = 3 × 3 × 5

The prime factors of 60 = 2 × 5 × 2 x 3

The common factors of 60 and 75 are 3 × 5.

Thus, HCF of (45, 60) = 15.

2.2 L-Division Method

This method is the most commonly used method and will be efficient in finding HCF of 2-digit / 3-digit numbers which has many factors.

Example Problem

Let us find the HCF of 24 and 15 using the L-Division method.

Step 1: Divide the given numbers by the least common prime factor.

Here, 3 is the least common prime factor of 24 and 15.

Step 2: Continue still there is no more common prime factor. Then multiply all the divisors.

Divisions of 24 and 15 by 3 will leave 8 and 5 as their remainders respectively. 8 and 5 do not have a common prime factor.

Hence, the HCF of 24 and 15 is 3.

2.3 Normal Division Method

This method is rarely used and will be more efficient in finding HCF of 3-digit / 4-digit odd numbers for which finding factors might look challenging.

Example Problem

Let's find the HCF of 24 and 15 by the normal division method.

Step 1: Divide the largest number by the smallest number.

Here, the largest number is 24 and the smaller one is 15. Divide 24 by 15

Step 2: Take the divisor as a new dividend and remainder as the new divisor, i.e. divide the first divisor by the first remainder.

Step 3: Proceed till the remainder is zero and the last divisor will be the HCF of the given numbers.

Therefore, HCF of 24 and 15 is 3.

3. Least Common Multiple (LCM)

The Lowest Common Multiple (LCM) of two or more numbers is the smallest of the common multiples of those numbers.

For example, let us take two numbers, 8 and 16.

Multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, and so on.

The multiples of 16 are 16, 32, 48, 64, 80, 96, and so on.

The first common value among these multiples is the Least Common Multiple (LCM) for 8 and 16, which is 16.

Thus, LCM of 8 & 16 is 16

4. How to Find LCM of the given numbers?

There are multiple methods to find the LCM of the given numbers. Prime factorisation method & L-Division method.

4.1. L-Division Method

This is the most commonly used method to find the LCM of the given numbers.

This method divides the given numbers by a common prime number until the remainder is a prime number or one. LCM will be the product obtained by multiplying all divisors and remaining prime numbers.

Example Problem

Find the LCM of 24 and 15 by the division method.

Solution:

Step 1: Divide the given numbers by the least prime number.

Here, 2 is the least number which will divide 24.

Step 2: Write the quotient and the number which is not divisible by the above prime number in the second row.

In the second row, write the quotient we get after the division of 24 by 2. Since 15 is not divisible by 2, write 15 in the second row as it is.

Step 3: Divide the numbers with another least prime number.

Step 4: Continue division until the remainder is a prime number or 1.

Step 5: Multiply all the divisors and the remaining prime number (if any) to obtain the LCM.

LCM of 24 and 15= 2×2×2×3×5=23×3×5

LCM of 24 and 15= 120

4.2 Prime Factorisation method

The following steps must be performed to find the LCM using the prime factorisation method. And this method is less commonly used.

Step 1: List the prime factors of the given numbers and note the common prime factors.

Step 2: The LCM of the given numbers = product of the common prime factors and the uncommon prime factors of the numbers.

Note: Common factors will be included only once.

Example Problem

Let us find the LCM of 45 and 60 using prime factorisation.

Step 1: The prime factors of 45 = 3 ×3 × 5 and 60 = 2 × 2 × 3 × 5

Step 2: The product of all the prime factors = Common prime factors (3 × 5) ×

Uncommon prime factors (2 × 2 × 3) = 180.

Therefore, LCM of 45 and 60 = 180.