Prime Time Class 6 Notes

Prime Time Class 6 Notes

Prime Time Class 6 Notes

1. Prime Numbers: Numbers that have only two factors, namely 1 and themselves. Examples include 2, 3, 5, 7,11, etc.

2. Composite Numbers: Numbers that have more than two factors, meaning they can be divided by at least one number other than 1 and themselves. Examples include 4, 6, 8, 9, etc.

3. Factors: A number is a factor of another number if it divides that number without leaving a remainder. For example, 4 is a factor of 12 because 12 ÷ 4 = 3.

4. Co-prime Numbers: Two numbers that do not have any common factors other than 1. For example, 8 and 9 are co-prime because their only common factor is 1.

5. Prime Factorization: The process of expressing a number as a product of its prime factors. For example, the prime factorization of 84 is 2 × 2 × 3 × 7.

6. Divisibility: A number is divisible by another if it can be divided by that number without leaving a remainder. For example, 8560 is divisible by 10.

7. Multiples: The result of multiplying a number by a positive integer. For example, the first few multiples of 10 are 10, 20, 30, etc.

8. Divisibility Rules: Guidelines that help determine if one number is divisible by another without performing long division. For example, a number is divisible by 5 if it ends in 0 or 5.

9. Rectangular Arrangement: The way of organizing a number of items into rows and columns such that the total number of items equals the product of the number of rows and columns.

Common Multiples and Common Factors

Common multiples are numbers that are multiples of two or more numbers. A multiple of a number is what you get when you multiply that number by an integer (like 1, 2, 3, etc.).

Example : Let’s say we have the numbers 3 and 4. The common multiples of 3 and 4 are :

12 (3 × 4 and 4 × 3)

24 (3 × 8 and 4 × 6)

So, the common multiples of 3 and 4 are 12, 24, and so on.

Common factors are numbers that can divide two or more numbers without leaving a remainder.

Example : Let’s take the numbers 12 and 18. The factors of 12 are :

1. 1, 2, 3, 4, 6, 12 The factors of 18 are :
2. 1, 2, 3, 6, 9, 18

Now, if we look at the lists of factors, we can see that the common factors of 12 and 18 are:

1. 1, 2, 3, and.6

So, the common factors of 12 and 18 are 1, 2, 3, and 6.

Prime Numbers

What are Prime Numbers?

Prime numbers are special numbers that have exactly two distinct positive divisors: 1 and the number itself. This means that a prime number can only be divided evenly (without leaving a remainder) by 1 and by itself.

Examples of Prime Numbers: The first few7 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on.

Important Note :

1. The number 1 is not a prime number because it only has one divisor (itself).
2. The number 2 is the only even prime number because all other even numbers can be divided by 2, which gives them more than two divisors.

Co-prime Numbers for Safekeeping

Treasures

What are Co-prime Numbers ?

Co-prime numbers (also known as relatively prime numbers) are two or more numbers that do not have any common factors other than 1. This means that the only number that can divide both of them evenly is 1.

Examples of Co-prime Numbers :

1. The numbers 8 and 15 are co-prime because their only common factor is 1.
2. The numbers 14 and 25 are also co-prime for the same reason.

Important Note : Any two prime numbers are always co-prime because they only have two factors : 1 and the number itself.

Prime Factorisation

What is Prime Factorisation?

Prime factorisation is the process of breaking down a number into its prime factors. Prime factors are the prime numbers that multiply together to give the original number. Every whole number greater than 1 can be expressed as a product of prime numbers in a unique way.

Why is Prime Factorisation Important?

1. It helps us understand the structure of numbers.
2. It is useful in simplifying fractions, finding the greatest common factor (GCF), and solving problems in number theory.

How to Find Prime Factorisation

Start with the Number: Choose a number you want to factor.

1. Divide by the Smallest Prime Number: Start dividing the number by the smallest prime number (which is 2) and continue dividing until you can’t divide anymore.
2. Repeat with the Resulting Quotient: If the quotient is not a prime number, repeat the process with the new number.
3. Write Down the Prime Factors: Once you can’t divide anymore, write down all the prime numbers you used.

Example : Finding the Prime Factorisation of 60

1. Start with 60.

2. Divide by 2 (the smallest prime number):

1. 60 ÷ 2 = 30

3. Divide 30 by 2 again :

1. 30 ÷ 2 = 15

4. Now, divide 15 by the next smallest prime number, which is 3 :

1. 15 ÷ 3 = 5

5. Finally, 5 is a prime number, so we stop here.

The prime factorisation of 60 is :

60 = 2 × 2 × 3 × 5or 60 = 2² × 3 × 5

Using Prime Factorisation to check if Two Numbers are Co-prime

How to check if Two Numbers are Co¬prime :

1. Find the Prime Factorisation of Both Numbers: For example, let’s check if 15 and, 28 are co-prime. Prime factorisation of 15 : 15 = 3 × 5. Prime factorisation of 28 : 28 = 2 × 2 × 7.
2. Look for Common Prime Factors: The prime factors of 15 are 3 and 5.The prime factors of 28 are 2 and 7.There are no common prime factors.

Conclusion: Since 15 and 28 do not share any prime factors, they are co-prime.

Using Prime Factorisation to check if One Number is Divisible by Another

A number A is divisible by another number B if you can divide A by B without leaving a remainder.

How to Check Divisibility Using Prime Factorisation :

1. Find the Prime Factorisation of Both Numbers: Let’s check if 30 is divisible by 6. Prime factorisation of 30 : 30 = 2 × 3 × 5. Prime factorisation of 6 : 6 = 2 × 3.
2. Check if the Prime Factors of 6 are in 30: The prime factors of 6 (2 and 3) are also in the prime factorisation of 30. Since all prime factors of 6 are included in the prime factorisation of 30, we can say that 30 is divisible by 6.

Conclusion : 30 is divisible by 6 because all the prime factors of 6 are present in the prime factorisation of 30.

Divisibility Tests

Common Divisibility Tests : Here are some common divisibility tests for the numbers 2, 5 and 10 :

1. Divisibility by 2 : A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8 (even numbers).

Example: 24 is divisible by 2 because it ends in 4.

2. Divisibility by 5 : A number is divisible by 5 if it ends in 0 or 5.

Example : 45 is divisible by 5 because it ends in 5.

3. Divisibility by 10: A number is divisible by 10 if it ends in 0.

Example: 150 is divisible by 10 because it ends in 0.