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CBSE Notes Class 6 Maths Chapter 5 Prime Time
Chapter 5: Prime Time in CBSE Class 6 Maths introduces students to the fascinating world of prime numbers, common factors, common multiples, and divisibility rules. It explores key concepts like prime factorisation, co-prime numbers, and divisibility tests, helping students understand the relationships between numbers and simplifying mathematical problems through these fundamental concepts.The accompanying PDF provides a convenient resource for revising the chapter and reviewing important concepts at a glance. It can be used for quick revision and practice alongside the textbook.
Common Factors
Factors are the numbers that can divide a given number completely. For example, the factors of 6 are 1, 2, 3, and 6 because these are the numbers that divide 6 without a remainder.
When we compare two or more numbers, the common factors are the numbers that divide both numbers completely.
Example: Common factors of 12 and 18 :
- The factors of 12 are 1, 2, 3, 4, 6, and 12.
- The factors of 18 are 1, 2, 3, 6, 9, and 18.
So, the common factors of 12 and 18 are 1, 2, 3, and 6. These are the numbers that can divide both 12 and 18 without leaving a remainder.
Common Multiples
A multiple is the result of multiplying a number by any whole number. For example, the multiples of 4 are 4, 8, 12, 16, 20, and so on.
Common multiples are the multiples that are shared by two or more numbers.
Example: Common Multiple of 4 and 6 :
- The multiples of 4 are 4, 8, 12, 16, 20, 24, and so on.
- The multiples of 6 are 6, 12, 18, 24, 30, and so on.
So, the common multiples of 4 and 6 are 12, 24, 36, and so on.
Note: The smallest common multiple of two numbers is called the Least Common Multiple or LCM.
Prime Numbers
A Prime Number is a number greater than 1 that has only two factors: 1 and the number itself. In other words, a prime number can only be divided by 1 and itself without leaving a remainder.
Here is a list of all prime numbers from 1 to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Co-prime Numbers
Two numbers are co-prime if their Greatest Common Factor (GCF) or Greatest Common Divisor (GCD) is 1. In other words, co-prime numbers don’t share any factors other than 1.
Example: 5 and 9, 8 and 15, 9 and 28, 5 and 14, 12 and 25, 7 and 30, etc
Prime Factorisation
Prime Factorisation is the process of breaking down a number into a product of its prime factors.
Examples:
- Prime Factorisation of 12 is 2 × 2 × 3.
- Prime Factorisation of 18 is 2 × 3 × 3.
Divisibility Tests
- A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- A number is divisible by 5 if its last digit is 0 or 5.
- A number is divisible by 6 if it is divisible by both 2 and 3.
- A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
- A number is divisible by 9 if the sum of its digits is divisible by 9.
- A number is divisible by 10 if its last digit is 0.
Key Concepts to Remember
- Factor: A whole number that divides a given number completely without leaving any remainder.
- Multiple: The product obtained when a given whole number is multiplied by any natural number (1, 2, 3, and so on).
- Properties of Factors: 1 is a factor of every number, every non-zero number is a factor of itself, every factor is less than or equal to the given number, and the total count of factors for any number is always finite.
- Properties of Multiples: Every number is a multiple of itself, every multiple is greater than or equal to the given number, and the total count of multiples for any number is infinite.
- Perfect Number: A number for which the sum of all its factors equals exactly twice the number itself, such as 6 and 28.
- Prime Number: A natural number greater than 1 that has exactly two unique factors, which are 1 and the number itself.
- Composite Number: A natural number greater than 1 that possesses more than two distinct factors.
- Special Status of One: The number 1 has only one factor (itself), making it neither a prime number nor a composite number.
- Even and Odd Primes: The number 2 is the smallest prime number and the only even prime number; all other prime numbers are odd.
- Twin Primes: A pair of prime numbers that have a numeric difference of exactly 2, such as 3 and 5, 5 and 7, 11 and 13, or 17 and 19.
- Co-Prime Numbers: Two numbers that share no common factors other than 1, such as 4 and 9 or 675 and 392.
- Prime Factorisation: Expressing a composite number as a unique product consisting solely of its prime factors.
- Highest Common Factor (HCF): The largest common factor that divides two or more given numbers without leaving a remainder.
- Least Common Multiple (LCM): The smallest non-zero common multiple shared by two or more given numbers.
1.0Important Formulas Prime Time
- Perfect Number Formula: A number N is perfect if Sumofallfactors=2N (e.g., for 6: 1+2+3+6=12=26).
- Co-Prime LCM Rule: If two numbers a and b are co-prime (HCF=1), then LCM(a,b)=ab.
HCF & LCM Bounds:
- HCF: Smallest number in the set.
- LCM: largest number in the set.
Divisibility Shortcuts:
- By 2, 5, 10: Check the last digit (2: ends in 0, 2, 4, 6, 8; 5: ends in 0 or 5; 10: ends in 0).
- By 3, 9: Check sum of digits (3: digit sum is divisible by 3; 9: digit sum is divisible by 9).
- By 4, 8: Check trailing digits (4: last 2 digits divisible by 4; 8: last 3 digits divisible by 8).
- By 6: Must satisfy divisibility tests for both 2 and 3.
Prime Testing Range Rules:
- For numbers 100–200: Test divisibility by primes less than 15 (2, 3, 5, 7, 11, 13).
- For numbers 200–400: Test divisibility by primes less than 20 (2, 3, 5, 7, 11, 13, 17, 19).
2.0Learning Guide for Class 6 Maths Prime Time - Chapter 5
- Exploring Factors with Square Tile Rectangles: Arrange square tiles into rectangular grids to discover all factors of a number. For example, 6 square tiles can be arranged as 16 or 23, showing that the factors of 6 are 1, 2, 3, and 6.
- Finding Common Factors with Venn Diagrams: List individual factors in separate circles; overlapping shared factors in the center intersection reveals the common factors. For 12 (factors: 1, 2, 3, 4, 6, 12) and 18 (factors: 1, 2, 3, 6, 9, 18), the shared intersection contains 1, 2, 3, and 6, making 6 the Highest Common Factor (HCF).
- Constructing the Sieve of Eratosthenes: Write numbers from 1 to 100. Cross out 1 (neither prime nor composite). Encircle 2 and cross out all its remaining multiples. Encircle the next uncrossed number 3 and cross out its multiples. Repeat for 5 and 7. All uncrossed encircled numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97) are the prime numbers up to 100.
- Prime Factorisation Using Factor Trees: Decompose a number into factor branches repeatedly until all endpoint nodes are prime numbers. For 108, branch into 254, then 54 into 227, 27 into 39, and 9 into 33, yielding 108=22333.
- Calculating HCF by Prime Factorisation: Write out prime factors for each number and multiply the lowest power of every common prime factor. For 98 (277) and 112 (22227), common prime factors are one 2 and one 7, giving HCF=27=14.
- Calculating LCM by Prime Factorisation: Write out prime factors for each number and multiply the highest power of all prime factors present. For 88 (2311) and 64 (26), take the highest powers 2611=6411=704, giving LCM=704.
- Solving Prime Product Grid Puzzles: Fill a grid with prime numbers so that row products match numbers on the right and column products match numbers at the bottom. For a row product of 154, fill the row with prime factors 2, 11, and 7 because 2117=154.
- Real-Life Application of Factors: Factors are used when dividing or sharing items into equal groups, such as sharing 12 chocolates equally among 3 friends (123=4).
- Real-Life Application of Multiples: Multiples are used in scheduling cyclic events, such as a bus arriving every 15 minutes (arriving at 12:15, 12:30, and 12:45 because 15, 30, and 45 are multiples of 15).
3.0Explore Chapter-wise Class 6 Maths Notes
4.0Additional Study Materials for Class 6 Maths
5.0Find the Right Study Resources for Class 6
6.0Benefits of CBSE Notes for Class 6 Maths Chapter 5 - Prime Time
- Mastering Prime Factorisation: You will learn how to break down any composite number into a product of its prime factors using methods like factor trees or division. This skill is crucial for finding the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM).
- Understanding Divisibility Tests: The CBSE Notes for Class 6 provides easy ways to check if a number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10 without performing actual division. This saves time and reduces errors. For instance, you'll learn that a number is divisible by 3 if the sum of its digits is divisible by 3 (e.g., for 123, 1+2+3=6, which is divisible by 3, so 123 is also divisible by 3).
- Preparation for Advanced Topics: The concepts learned in this chapter form a strong foundation for more advanced topics in mathematics that you will encounter in higher classes.
- Quick Revision: CBSE Notes provide a concise summary of the chapter, making it easy to revise the key concepts and formulas before tests or exams.
- Improved Problem-Solving Skills: A clear understanding of the concepts and techniques discussed in the notes will enhance your ability to solve a variety of problems related to factors, multiples, and prime numbers.