Concise Mathematics Class 6 ICSE Solutions for Chapter 8 - HCF and LCM

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Let’s have a look at the Concept of ICSE Mathematics class 6 Solutions Chapter 8

Prime Numbers: It is a natural number, which is divisible by only one, and itself is called a prime number.

Highest Common Factor: H.C.F. stands for Highest Common Factor. The H.C.F. of two or more given numbers is the greatest number (also known as a factor) which divides each given number completely.

Lowest Common Factor: L.C.M. stands for Lowest Common Multiple. The L.C.M. of two or more given numbers is the lowest (also known as smallest) number which is exactly divisible by each of the given numbers.

To find H.C.F. and L.C.M. of numbers we have two important methods that are the Prime factorization method and the division method. The shortcut method to find both H.C.F. and L.C.M. is a division method. We will also solve a few problems based on these two concepts to understand them better.

HCF and LCM Definition

HCF (Highest Common Factor)

The greatest common divisor or the gcd of two or more positive integers is the greatest positive integer that divides the numbers without leaving a remainder. For example, take 8 and 12. The H.C.F. of 8 and 12 will be 4 because the highest number that can divide both the numbers is 4.

LCM (Least Common Multiple)

The least common multiple or LCM of two numbers i.e a and b is denoted as LCM (a,b).  The LCM is the smallest or least positive integer which is divisible by both the numbers. For example, Suppose two positive integers are 4 and 6.

Multiples of 4 are: 4,8,12,16,20,24…

Multiples of 6 are: 6,12,18,24….

As we can see, 12 is a common multiple. So, 12 is the LCM of 4 and 6.

List of HCF and LCM Properties

Property 1: The product of LCM and HCF of any two given natural numbers is equal to the product of those two given numbers.

Proof: Here, we need to prove  LCM × HCF = Product of the Numbers

Suppose A and B are two numbers, then.

LCM (A and B) × HCF (A and B) = A × B

For example: If 3 and 8 are two numbers, and LCM (3, 8) = 24, HCF (3, 8) = 1

Sol: LCM (3, 8) x HCF (3, 8) = 24 x 1 = 24

Also, 3 x 8 = 24

Hence, proved.

Note: We can apply this property only for two numbers.

Property 2: HCF of co-prime numbers is 1. Therefore, the LCM of co-prime numbers is equal to the product of the numbers.

Proof: We need to prove LCM of Co-prime Numbers = Product of The Numbers.

For 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 = (LCM of numerators)/(HCF  of denominators)

HCF of fractions = (HCF of numerators)/(LCM of denominators)

For example: Let us take two fractions 4/9 and 6/21.

4 and 6 are the numerators whereas 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 will never be greater than the given numbers.

For example, the HCF of 4 and 8 is 4.

Here, one number is 4  and another number is 8 and HCF is also 4, which is greater than the HCF (4, 8).

Property 5: LCM of any two or more numbers will never be smaller than the given numbers.

For example, the LCM of 4 and 8 is 8 which is not smaller than any of them.

Solved Examples:

1. Prove that: LCM (9 and 12) × HCF (9 &=and 12) = Product of 9 and 12.

Ans: Given number is 9 and 12. We need to find factors 9 and 12.

Factor of 9 = 3 × 3 = 32

Factor of 12 = 2 × 2 × 3 = 22 × 3

LCM of 9 and 12 = 22 × 32 = 4 × 9 = 36

HCF of 9 and 12 = 3

We know that HCF and LCM of two numbers = Product of that numbers

LCM (9 and 12) × HCF (9 and 12) = 36 × 3 = 108

Product of 9 and 12 is 108

Hence, LCM (9 and 12) × HCF (9 and 12) = 9 × 12 = 108. 

Hence Proved.

2. Find the highest common factor of 25, 35, and 45 by the prime factorisation method.

Ans: Given three numbers as 25, 35, and 45.

We know, 25 = 5 × 5

35 = 5 × 7

45 = 5 × 9

From the above expression, we can see 5 is the only common factor for all three numbers.

Therefore, the HCF of 25, 35, and 45 is 5.

Conclusion

Here we have discussed ICSE maths class 6 chapter 8 solutions. So, prime factorization is the process of breaking up a number into factors until only prime factors are left.

We have also learned that, 

The Highest Common Factor (or HCF) of two or more given numbers is the highest common factor.

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

This solution is very important because we have covered all the important concepts of ICSE Mathematics Class 6 Solutions Chapter 8.