ICSE Class 7 Maths Chapter 03 HCF LCM

Chapter 3: H.C.F. and L.C.M.

3.1 Elementary Treatment

1. Factors

Each of the natural numbers that divides a given number exactly (completely) is called a factor of the given number.

For example:

(a) Each of the natural numbers 1, 2, 3 and 6 divides the number 6 completely; therefore each of 1, 2, 3 and 6 is a factor of 6.

(b) Each of the natural numbers 1, 2, 4, 8 and 16 divides the number 16 exactly; therefore 1, 2, 4, 8 and 16 are factors of 16.

For the same reason:

(c) Factors of 7 are: 1 and 7

(d) Factors of 35 are: 1, 5, 7 and 35

(e) Factors of 20 are: 1, 2, 4, 5, 10 and 20 and so on.

Moreover:

Since, 7 divides 56 exactly; 7 is called a factor (or divisor) of 56 and 56 is called a multiple of 7.

Similarly, 8 divides 72 exactly; so 8 is a factor of 72 whereas 72 is a multiple of 8.

Remember:

(a) 1 (one) is a factor of every number.

(b) Every non-zero number is factor as well as multiple of itself. For example, number 5 is a factor of itself as 5 divides 5 completely and 5 is also a multiple of 5.

(c) Except one (1), every natural number has at least two factors. For example, (i) Factors of 2 are: 1 and 2, (ii) Factors of 3 are: 1 and 3, (iii) Factors of 4 are: 1, 2 and 4 and so on.

(d) Every natural number has an infinity number of multiples. For example, (i) Multiples of 1 are: 1, 2, 3, 4, 5, ...... (ii) Multiples of 2 are: 2, 4, 6, 8, 10, ...... (iii) Multiples of 3 are: 3, 6, 9, 12, ......

Key Points:

1. Every prime number has two factors, i.e., 1 (unity) and the number itself. For example, (i) 5 has only two factors: 1 and 5, (ii) 17 has only two factors: 1 and 17 and so on.

2. Two (2) is the only prime number which is even also.

3. Every number, which is greater than 1 and is not prime, is called a composite number and every composite number has three or more factors. For example, (i) 4 is a composite number as it has three factors: 1, 2 and 4. (ii) since, 10 has four factors: 1, 2, 5 and 10; 10 is a composite number and so on.

4. The number 1 (unity) has only one factor, i.e., 1 itself. So, the number one (1) is neither prime nor composite.

Teacher's Note

Understanding factors is essential for everyday tasks like dividing objects equally or calculating portions in recipes and construction projects.

2. Common factors

A common factor of two or more numbers is a number which divides each of the given numbers exactly.

For example:

(i) Factors of 6 = 1, 2, 3 and 6 and factors of 8 = 1, 2, 4 and 8. Common factors of 6 and 8 = 1 and 2. It can easily be shown that each common factor (1 and 2) divides the given numbers 6 and 8 exactly.

(ii) Factors of 8 = 1, 2, 4 and 8, factors of 12 = 1, 2, 3, 4, 6 and 12 and factors of 16 = 1, 2, 4, 8 and 16. Common factors of 8, 12 and 16 = 1, 2 and 4. [Check whether each common factor divides each given number (8, 12 and 16) exactly or not].

Example 1:

Find the common factors of:

(i) 10 and 15

(ii) 14, 21 and 42

Solution:

(i) Factors of 10 = 1, 2, 5 and 10 and, factors of 15 = 1, 3, 5 and 15. Common factors of 10 and 15 = 1 and 5 (Ans.)

(ii) Factors of 14 = 1, 2, 7 and 14. Factors of 21 = 1, 3, 7 and 21 and, factors of 42 = 1, 2, 3, 6, 7, 14, 21 and 42. Common factors of 14, 21 and 42 = 1 and 7 (Ans.)

Teacher's Note

Common factors help us find the greatest measure for sharing items, such as grouping students into equal teams or distributing supplies efficiently.

3. Prime factor

When a factor of a given number is a prime number also, it is called a prime factor of the given number.

Since, 3 is a factor of 12 and 3 is a prime number also; therefore 3 is a prime factor of 12.

In the same way.

(i) Factors of 14 are 1, 2, 7 and 14, out of these 2 and 7 are prime numbers; therefore, 2 and 7 are prime factors of 14.

(ii) Factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42; out of these factors 2, 3 and 7 are prime numbers also, therefore prime factors of 42 are 2, 3 and 7.

4. Repeated prime factors

Since, 8 = 2 x 2 x 2; 2 is said to be repeated prime factor of 8.

A number can have two or more different repeated prime factors.

(i) 36 = 2 x 2 x 3 x 3

(ii) 108 = 2 x 2 x 3 x 3 x 3

(iii) 225 = 3 x 3 x 5 x 5 and so on

3.2 H.C.F. (Highest Common Factor)

Highest common factor (H.C.F.) of two or more given numbers is the greatest number which divides each of the given numbers exactly.

For example:

H.C.F. of 18, 24 and 36 is 6, as 6 is the greatest number which divides each of 18, 24 and 36 exactly.

In other way:

Factors of 18 = 1, 2, 3, 6, 9 and 18, factors of 24 = 1, 2, 3, 4, 6, 8, 12 and 24 and, factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18 and 36. Common factors of 18, 24 and 36 = 1, 2, 3 and 6. Highest common factor of 18, 24 and 36 = 6.

3.3 Methods of Finding H.C.F.

1. Common factor method:

Steps: 1. Find all the factors of each given numbers. 2. Find common factors of the given numbers. 3. The greatest of all the factors obtained in Step 2, is the required H.C.F.

Example 2:

Find, using common factor method, H.C.F. of 27, 54 and 81

Solution:

Factors of 27 = 1, 3, 9 and 27; factors of 54 = 1, 2, 3, 6, 9, 18, 27 and 54 and, factors of 81 = 1, 3, 9, 27 and 81. Common factors of 27, 54 and 81 = 1, 3, 9 and 27. Required H.C.F. = 27 (Ans.)

Teacher's Note

Finding the highest common factor is useful in real life for simplifying fractions, creating equal groups, and solving problems involving proportions.

2. Prime factor method

Steps: 1. Express each given number as the product of its prime factors. 2. From the result of step 1, find out all the prime factors which are common and then multiply these common prime factors to get the required H.C.F.

Example 3:

Find, using prime factor method, the H.C.F. of:

(i) 84 and 105

(ii) 124, 296 and 228

Solution:

(i) Step 1: 84 = 2 x 2 x 3 x 7 and 105 = 3 x 5 x 7. Step 2: H.C.F. = 3 x 7 = 21 (Ans.)

(ii) Step 1: 124 = 2 x 2 x 31, 296 = 2 x 2 x 2 x 37 and 228 = 2 x 2 x 3 x 19. Step 2: H.C.F. = 2 x 2 = 4 (Ans.)

Alternative method:

(i) Step 1: Express each given number as the product of its prime factors and then in the exponent form. Step 2: Then required H.C.F. is the product of all the common prime factors with lowest powers of them.

(ii) Step 1: 124 = 2 x 2 x 31 = \(2^2\) x 31, 296 = 2 x 2 x 2 x 37 = \(2^3\) x 37 and, 228 = 2 x 2 x 3 x 19 = \(2^2\) x 3 x 19. Step 2: Since, common prime factor with lowest power of it is \(2^2\). H.C.F. = \(2^2\) = 2 x 2 = 4 (Ans.)

3. Division method:

Steps: 1. Divide the greater number by the smaller number. 2. By the remainder of division in Step 1; divide the smaller number. 3. By the remainder in Step 2, divide the remainder of Step 1. 4. Continue in the same way, till no remainder is left. The last divisor is the required H.C.F.

Example 4:

Find, using division method, the H.C.F. of:

(i) 180 and 270

(ii) 852 and 1065

Solution:

(i) Step 1: 180 ) 270 ( 1. Remainder 90. Step 2: 90 ) 180 ( 2. Remainder 0. H.C.F. = 90 (Ans.)

(ii) 852 ) 1065 ( 1. Remainder 213. 213 ) 852 ( 4. Remainder 0. H.C.F. = 213 (Ans.)

Example 5:

Find, using division method, the H.C.F. of:

(i) 18 and 30

(ii) 75 and 180

Solution:

(i) Step 1: 18 ) 30 ( 1. Remainder 12. Step 2: 12 ) 18 ( 1. Remainder 6. Step 3: 6 ) 12 ( 2. No remainder. H.C.F. = 6 (Ans.)

(ii) 75 ) 180 ( 2. Remainder 30. 30 ) 75 ( 2. Remainder 15. 15 ) 30 ( 2. No remainder. H.C.F. = 15 (Ans.)

Teacher's Note

The division method for finding H.C.F. is like the process of measuring lengths with different rulers - you keep dividing until you find the largest measurement that fits exactly into all the original lengths.

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