Properties Of HCF And LCM
Properties of HCF and LCM: For a better understanding of the concepts of LCM (Lowest Common Multiple) and HCF (Highest Common Factor), we need to recollect the terms multiples and factors. Let’s learn about LCM, HCF, and the relation between HCF and LCM of natural numbers.
Learn in detail: Hcf And Lcm
Definition of LCM and HCF
Lowest Common Multiple (LCM): The least or smallest common multiple of any two or more given natural numbers are termed as LCM. For example, LCM of 10, 15, and 20 is 60.
Highest Common Factor (HCF): The largest or greatest factor common to any two or more given natural numbers is termed as HCF of given numbers. Also known as GCD (Greatest Common Divisor). For example, HCF of 4, 6 and 8 is 2.
4 = 2 × 2
6 =3 × 2
8 = 4 × 2
Here, the highest common factor of 4, 6 and 8 is 2.
LCM and HCF Formula
We can write the formula for LCM and HCF in terms of the sentence to get the LCM and HCF for any given set of numbers quickly, considering their prime factorisation.
LCM = Product of the greatest power of each prime factor, involved in the numbers
HCF = Product of the smallest power of each common prime factor in the numbers
Example:
Let us consider two numbers, 8 and 20.
The prime factorisation of 8 and 20 can be written as:
8 = 2 × 2 × 2 = 2^{3}
20 = 2 × 2 × 5 = 2^{2} × 5
Now,
LCM = 2^{3} × 5 = 8 × 5 = 40
HCF = 2^{2} = 4
This way of finding LCM and HCF of numbers is often referred to as one of the methods to find HCF and LCM, i.e. the Prime factorisation Method.
In the previous example, LCM and HCF are calculated by listing the common factors of numbers. We can also, find the HCF of given numbers using the long division method. However, HCF and LCM of given numbers can be found by using two methods; they are division method and prime factorisation.
List of HCF and LCM Properties
Property 1
The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.
LCM × HCF = Product of the Numbers
Suppose A and B are two numbers, then.
LCM (A & B) × HCF (A & B) = A × B
Example: If 3 and 8 are two numbers.
LCM (3,8) = 24
HCF (3,8) = 1
LCM (3,8) x HCF (3,8) = 24 x 1 = 24
Also, 3 x 8 = 24
Hence, proved.
Note: This property is applicable for only two numbers.
Property 2
HCF of coprime numbers is 1. Therefore, the LCM of given coprime numbers is equal to the product of the numbers.
LCM of Coprime Numbers = Product Of The Numbers
Example: Let us take two coprime numbers, such as 21 and 22.
LCM of 21 and 22 = 462
Product of 21 and 22 = 462
LCM (21, 22) = 21 x 22
Property 3
H.C.F. and L.C.M. of Fractions:
LCM of fractions = \(\frac{LCM \: of\: numerators}{HCF \: of\: denominators}\)
HCF of fractions = \(\frac{HCF \: of\: numerators}{LCM \: of\: denominators}\)
Example: Let us take two fractions 4/9 and 6/21.
4 and 6 are the numerators & 9 and 12 are the denominators
LCM (4, 6) = 12
HCF (4, 6) = 2
LCM (9, 21) = 63
HCF (9, 21) = 3
Now as per the formula, we can write:
LCM (4/9, 6/21) = 12/3 = 4
HCF (4/9, 6/21) = 2/63
Property 4
HCF of any two or more numbers is never greater than any of the given numbers.
Example: HCF of 4 and 8 is 4
Here, one number is 4 itself and another number 8 is greater than HCF (4, 8), i.e.,4.
Property 5
LCM of any two or more numbers is never smaller than any of the given numbers.
Example: LCM of 4 and 8 is 8 which is not smaller to any of them.
Solved Problems
Example 1: Prove that: LCM (9 & 12) × HCF (9 & 12) = Product of 9 and 12
Solution:
9 = 3 × 3 = 3²
12 = 2 × 2 × 3 = 2² × 3
LCM of 9 and 12 = 2² × 3² = 4 × 9 = 36
HCF of 9 and 12 = 3
LCM (9 & 12) × HCF (9 & 12) = 36 × 3 = 108
Product of 9 and 12 = 9 × 12 = 108
Hence, LCM (9 & 12) × HCF (9 & 12) = 9 × 12 = 108. Proved.
Example 2: 8 and 9 are two coprime numbers. Using these numbers verify, LCM of Coprime Numbers = Product Of The Numbers.
Solution: LCM and HCF of 8 and 9:
8 = 2 × 2 × 2 = 2³
9 = 3 × 3 = 3²
LCM of 8 and 9 = 2³ × 3² = 8 × 9 = 72
HCF of 8 and 9 = 1
Product of 8 and 9 = 8 × 9 = 72
Hence, LCM of coprime numbers = Product of the numbers. Therefore, verified.
Example 3: Find the HCF of \(\frac{12}{25}\), \(\frac{9}{10}\), \(\frac{18}{35}\), \(\frac{21}{40}\).
Solution: Solution:
12 = 2 × 2 × 3
9 = 3 × 3
18 = 2 × 3 × 3
21 = 3 × 7
HCF (12, 9, 18, 21) = 3
25 = 5 × 5
10 = 2 × 5
35 = 5 × 7
40 = 2 × 2 × 2 × 5
LCM(25, 10, 35, 40) = 5 × 5 × 2 × 2 × 2 × 7 = 1400
The required HCF = HCF(12, 9, 18, 21)/LCM(25, 10, 35, 40) = 3/1400
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