Maharashtra Board Class 7 Maths part 1 Chapter 3 HCF and LCM PDF Download

HCF And LCM

Let's Recall

Which is the smallest prime number?

List the prime numbers from 1 to 50. How many are they?

Circle the prime numbers in the list below.

17, 15, 4, 3, 1, 2, 12, 23, 27, 35, 41, 43, 58, 51, 72, 79, 91, 97

Co-Prime Numbers

Two numbers which have only 1 as a common factor are said to be co-prime or relatively prime or mutually prime numbers.

For example, 10 and 21 are co-primes, because the divisors of 10 are 1, 2, 5, 10 while the divisors of 21 are 1, 3, 7, 21 and the only factor common to both 10 and 21 is 1.

Some other co-prime numbers are (3, 8); (4, 9); (21, 22); (22, 23); (23, 24).

Verify that any two consecutive natural numbers are co-primes.

Teacher's Note

Co-prime numbers are two numbers that have no common factor except 1. For example, 7 and 9 are co-primes because they share only 1 as a factor, like how your two different schools have no common friend.

Exam Trick

Remember: Any two numbers that come one after another (like 5 and 6 or 10 and 11) are always co-primes. This is the easiest pair to remember in exams.

Points to Remember

Co-prime numbers have only 1 as a common factor.

Two consecutive numbers are always co-primes.

Example: 3 and 4 are co-primes.

Twin Prime Numbers

If the difference between two co-prime numbers is 2, the numbers are said to be twin prime numbers.

For example: (3, 5); (5, 7); (11, 13); (29, 31) etc.

Teacher's Note

Twin primes are two prime numbers that differ by exactly 2. Like twin babies are born together, twin primes are always 2 apart, such as 11 and 13.

Exam Trick

Difference = 2 is the key. If two primes are 2 apart, they are twins. Just remember 3 and 5, which is the smallest twin prime pair.

Points to Remember

Twin primes have a difference of exactly 2.

Both numbers must be prime.

Examples: (3, 5), (5, 7), (11, 13), (29, 31).

Practice Set 10

1. Which number is neither a prime number nor a composite number?

2. Which of the following are pairs of co-primes?

(i) 8, 14 (ii) 4, 5 (iii) 17, 19 (iv) 27, 15

3. List the prime numbers from 25 to 100 and say how many they are.

4. Write all the twin prime numbers from 51 to 100.

5. Write 5 pairs of twin prime numbers from 1 to 50.

6. Which are the even prime numbers?

Factorising A Number Into Its Prime Factors

A simple but important rule given by Euclid is often used to find the GCD or HCF and LCM of numbers. The rule says that any composite number can be written as the product of prime numbers.

Teacher's Note

Prime factorisation means breaking a number into smaller prime numbers. For example, 12 = 2 × 2 × 3. This is like breaking a big chocolate bar into smaller pieces.

Exam Trick

Always divide by the smallest prime first (2, then 3, then 5, and so on). Keep dividing until you get 1. This method never fails.

Points to Remember

Every composite number is a product of prime numbers.

Start dividing by 2, then 3, then 5, and so on.

Continue until you reach 1.

Let Us Learn How To Find The Prime Factors Of A Number

Example: Write the number 24 in the form of the product of its prime factors.

Method for finding prime factors

Vertical Arrangement

2 | 24

2 | 12

2 | 6

3 | 3

| 1

24 = 2 × 2 × 2 × 3

Horizontal Arrangement

24 = 2 × 12

= 2 × 2 × 6 ...12 is factorised.

= 2 × 2 × 2 × 3 ...6 is factorised.

The prime factors are 2 and 3.

Remember: To write a given number as a product of its prime factors is to factorise it into primes.

Example: Write each of the given numbers as a product of its prime factors.

63

7 × 9

7 × 3 × 3

63 = 7 × 3 × 3

45

5 × 9

5 × 3 × 3

45 = 5 × 3 × 3

20

2 × 10

2 × 2 × 5

20 = 2 × 2 × 5

Example: Factorise into primes: 117.

3 | 117

3 | 39

13 | 13

| 1

117 = 3 × 3 × 13

Example: Factorise into primes: 250.

2 | 250

5 | 125

5 | 25

5 | 5

| 1

250 = 2 × 5 × 5 × 5

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