WBBSE Notes For Class 6 Maths Chapter 1 Simplification Prime And Composite Numbers

Chapter 1 Simplification Prime And Composite Numbers

Chapter 1 Prime And Composite Numbers Definition: 

A number is said to be a prime number if it is divisible either by itself or by 1 only and is not divisible by any other number.

2, 3, 5, 7, 11, etc. are prime numbers. A prime number has no factor except the. the number itself and 1.

Definition:

1. A number that is divisible by numbers other than itself and 1 is called a composite number.
2. For example, 4, 12, 15, 18, 24, etc., numbers are composite numbers. A composite number is a multiple of two or more prime numbers.
3. For example, 6 is a composite number and it is a multiple of the prime numbers 2 and 3.
4. Again 12 is a composite number and it is a multiple of 2, 3, 4, and 6.
5. Among them, 2 and 3 are the prime numbers, and 4, and 6 are composite numbers.
6. Here 2, 3, 4, and 6 are the factors of 12.
7. The number 1 is not a prime number, it is also not a composite number. 

Read And Learn More: WBBSE Notes For Class 6 Maths Chapter 1 Simplification

Determination of Prime Numbers :

1. The method of determination of prime numbers among the natural numbers 1, 2, 3, 4, 5, 6, etc., is given below:
2. First, we write down the natural numbers 1, 2, 3, 4, 5, 6, etc. consecutively.
3. Then cut out every second number after 2. Thus all the multiples of 2 are canceled.
4. Then cancel every third number after 3 and so all the multiples of 3 are canceled.
5. Now cancel every fifth number after 5. Following the same procedure as the prime numbers 7, 11, etc., the numbers which are left after cancellation are the prime numbers.
6. With the help of the above process, the prime numbers from 1 to 50 have been determined as follows

 

WBBSE Class 6 Prime and Composite Numbers Notes

1. ∴ The remaining numbers are 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
2. These numbers are prime numbers except 1.
3. Therefore the prime numbers between 1 and 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
4. This method is called the Sieve of Eratosthenes. In third century B.
5. C., Greek Mathematician Eratosthenes formulated a method by which prime numbers between 1 to 100 could be identified.
6. In this method, the prime numbers can be easily found.
7. out without finding factors or multiples.

 

Numbers Prime to each other or Co-prime numbers :

Definition:

1. If two numbers are such that they do not have any common factors, or they have only common factor 1, then the numbers are said to be prime to each other (or Co-príme numbers).
2. In the two numbers which are prime to each other or Co-prime numbers, there is no common factor other than 1.
3. These numbers themselves may or may not be prime.

Important Definitions Related to Prime and Composite Numbers

For example

1. 11 and 19 both the numbers are both prime and they do not have any common factor other than 1.
2. Therefore 11 and 19 are prime to each other.
3. Similarly, 11 and 13 are prime to each other because they do not have any common factor other than 1.
4. Again 12 = 2 x 2 x 3 and 35 = 5 x 7. So the numbers 12 and 35 both are composite numbers i.e., they are not prime numbers.
5. But there is no number common other than 1, by which both of them are exactly divisible i.e., they have no common factor.
6. So the numbers 12 and 35 are prime to each other although they are not prime numbers.

 

Factors and Prime Factors:

1. We know that,12 = 2 x 2 x 3. We can write,
2. \(\left.\begin{array}{rl}
   12 & =1 \times 12 \\
   & =2 \times 6 \\
   & =3 \times 4
   \end{array}\right\}\)
3. ∴ The factors of 12 are 1, 2, 3, 4, 6, and 12. 
4. Among them, 2 and 3 are prime factors of 12. 
5. So the prime factors of 12 are 2 and 3.
6. Similarly, 
7. \(\left.35=\begin{array}{c}
   1 \times 35 \\
   5 \times 7
   \end{array}\right\}\)
8. ∴ The factors of 35 are 1, 5, 7, and 35. 
9. Among them, 5 and 7 are prime factors.
10. ∴ The prime factors of 35 are 5 and 7.

Simplification Maths Class 6

Definition:

1. The numbers by which a given number is exactly divisible, then the numbers are called the factors of the given number.
2. Among these factors, which factors are prime numbers are called prime factors.
3. So the factors and prime factors of a given number are not the same.

Resolution of a number into Prime factors:

1. In order to resolve a number into prime factors, the number should be continuously divided by suitable prime numbers, until the quotient comes to be a prime number.
2. The successive divisors and the last quotient will be the prime factors.

Understanding Prime Numbers

Rule of Divisibility:

1. The following are the rules to determine whether a natural number is divisible by other natural numbers or not.
2. The natural number which has 0 or an even number in the unit’s place is divisible by 2.
   So all natural numbers having 0 or an even number in the unit’s place is divisible by 2. For example, 20, 26, 32, 34, etc. are divisible by 2.
3. If the sum of the digits of a natural number is divisible by 3, the number is also divisible by 3.
   For example, the sum of the digits of the number 234 is 9 which is divisible by 3 and so the number 234 is divisible by 3.
4. If the last two digits of a natural number be zeroes or if the number formed by the last two digits of the given natural number is divisible by 4, then the given natural number is divisible by 4.
   For example, the natural number 200, having the last two digits 0, is divisible by 4 and the number formed by the last two digits of the natural number 132 is 32, which is divisible by 4 and so the number 132 is divisible by 4.
5. The natural numbers having 0 or 5 in the unit’s place are divisible by 5.
   For example, 150 and 205 are divisible by 5.
6. The natural numbers which are divisible by both 2 and 3 are divisible by 6. For example, 78 and 126 are divisible by 6.
7. In order to determine whether a natural number is divisible by 7 or not, it is better to divide the natural number by 7.
   If there is no remainder, then the given natural number must be divisible by 7.
8. If the last 3 digits of a given natural number be zeroes or the number formed by the last 3 digits of the given natural number is divisible by 8, then the given natural number must be divisible by 8.
   For example, 1000 and 2152 both are divisible.. by 8.
9. If the sum, of the digits of a number, is divisible by 9, then the number is divisible by 9.
   For example, the numbers 1107 and 1827 are divisible by 9.
10. The natural numbers having 0 in the unit’s place must be divisible by 10.
    For example, 210, and 2350 both are divisible by 10.
11. If the difference of the sum of the digits in the odd places and even places of a natural number by zero or divisible by 11, then the number must be divisible by 11.
    For example, the numbers 1474 and 61754 are divisible by 11, because for the number 1474:
12. The sum of the digits in the odd places = 1 + 7 = 8 and the sum of the digits in the even places 4+4 = 8.
13. The difference between these two numbers = is 8-8 = 0.
14. Again for the number 61754:
15. The sum of the digits in the odd places = 6 +7+4 = 17 and the sum of the digits in the even places = 1 + 5 = 6.
16. ∴ The difference between these two numbers is 17 – 6 = 11, which is divisible by 11.
17. Hence both the natural numbers 1474 and 61754 are divisible by 11.
18. With the help of the above rules, we can determine whether a natural number is divisible, by 12, 15, 16, 18, 25, etc., or not.