Least Common Multiple

LCM stands for Least Common Multiple, which is the smallest multiple that two or more numbers have in common. For example, the LCM of 2 and 3 is 6 because 6 is the smallest number that is a multiple of both 2 and 3. Similarly, the LCM of 4, 6, and 8 is 24 because it is the smallest number that is a multiple of all three numbers.

Highest Common Factor

HCF stands for Highest Common Factor, which is the largest factor that two or more numbers have in common. For example, the HCF of 12 and 18 is 6 because 6 is the largest factor that both numbers share. Similarly, the HCF of 16 and 24 is 8 because it is the largest factor that both numbers share.

Relation between LCM and HCF

The relationship between LCM (Least Common Multiple) and HCF (Highest Common Factor) is important to understand in number theory. LCM and HCF are related by the following formula:

LCM × HCF = Product of the two numbers

This means that if we have two numbers A and B, their product (AB) is equal to the product of their LCM and HCF. Therefore, if we know the LCM and HCF of two numbers, we can calculate their product. Similarly, if we know the product of two numbers and one of their factors (e.g. HCF), we can calculate the other factor (e.g. LCM).

For example, if the LCM of two numbers is 60 and their HCF is 3, we can find the numbers using the formula:

Product of the two numbers = LCM × HCF = 60 × 3 = 180

Now, we need to find two numbers whose product is 180 and HCF is 3. We can list the factors of 180 (1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180) and identify the pair whose HCF is 3 (i.e. 9 and 20). Therefore, the two numbers are 9 and 20.

In summary, the relationship between LCM and HCF is crucial in solving problems related to number theory, and understanding this relationship can help you solve problems more efficiently.

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Example 1: Find the LCM and HCF of 12 and 20.

Solution:
To find the LCM, we need to list the multiples of 12 and 20 until we find the smallest number that is common to both lists.
Multiples of 12: 12, 24, 36, 48, 60, 72, ...
Multiples of 20: 20, 40, 60, 80, 100, ...
So, the LCM of 12 and 20 is 60.

To find the HCF, we need to find the highest common factor that divides both numbers.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 20: 1, 2, 4, 5, 10, 20
So, the HCF of 12 and 20 is 4.

Answer:
LCM = 60
HCF = 4

Example 2: Find the LCM and HCF of 15 and 25.

Solution:
Multiples of 15: 15, 30, 45, 60, 75, ...
Multiples of 25: 25, 50, 75, 100, ...
So, the LCM of 15 and 25 is 75.

Factors of 15: 1, 3, 5, 15
Factors of 25: 1, 5, 25
So, the HCF of 15 and 25 is 5.

Answer:
LCM = 75
HCF = 5

Example 3: Find the LCM and HCF of 36 and 48.

Solution:
Multiples of 36: 36, 72, 108, 144, ...
Multiples of 48: 48, 96, 144, 192, ...
So, the LCM of 36 and 48 is 144.

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
So, the HCF of 36 and 48 is 12.

Answer:
LCM = 144
HCF = 12

LCM and HCF for Sainik School Entrance Examination