CBSE Class 6 Maths: Playing with Numbers Exercise 3.2 - Even, Odd, Prime & Composite

What are Even and Odd Numbers?

Numbers are broadly classified into two main types based on their divisibility by 2: Even Numbers and Odd Numbers. This is one of the most fundamental classifications you'll encounter in mathematics.

Even Numbers: A number that is completely divisible by 2 is called an even number. This means when you divide an even number by 2, the remainder is always 0. Another easy way to spot even numbers is by looking at their last digit. If a number ends with 0, 2, 4, 6, or 8, it is an even number. For example, 2, 4, 6, 8, 10, 12, 50, 100, and even 12,34,568 are all even numbers.

Odd Numbers: A number that is not completely divisible by 2 is called an odd number. When you divide an odd number by 2, the remainder is always 1. Similar to even numbers, you can identify odd numbers by their last digit. If a number ends with 1, 3, 5, 7, or 9, it is an odd number. For example, 1, 3, 5, 7, 9, 11, 23, 75, and 98,76,543 are all odd numbers. It's important to remember that every whole number is either even or odd, there's no third option!

Prime and Composite Numbers - The Building Blocks

Beyond even and odd, numbers also have another fascinating classification based on their factors: Prime and Composite Numbers. Factors are numbers that divide a given number exactly, leaving no remainder.

Prime Numbers: A prime number is a whole number greater than 1 that has exactly two factors: 1 and the number itself. Think of them as the fundamental building blocks for all other numbers through multiplication. For example:

• 2 is a prime number (factors: 1, 2). It's the smallest prime number and the only even prime number.
• 3 is a prime number (factors: 1, 3).
• 5 is a prime number (factors: 1, 5).
• 7 is a prime number (factors: 1, 7).

Composite Numbers: A composite number is a whole number greater than 1 that has more than two factors. These numbers can be formed by multiplying smaller prime numbers. For example:

• 4 is a composite number (factors: 1, 2, 4). It's the smallest composite number.
• 6 is a composite number (factors: 1, 2, 3, 6).
• 9 is a composite number (factors: 1, 3, 9).
• 12 is a composite number (factors: 1, 2, 3, 4, 6, 12).

What about the number 1? The number 1 is special! It has only one factor (which is 1 itself). Because it doesn't have exactly two factors (like primes) or more than two factors (like composites), 1 is neither a prime nor a composite number. This is a very important point to remember in your class 6 maths journey!

How to Identify Prime and Composite Numbers (Step-by-Step)

1. Step 1: Check if the number is 1 — If the number is 1, it is neither prime nor composite. You are done!
2. Step 2: Find all factors of the number — Start by dividing the number by 1, then by 2, then by 3, and so on, up to the number itself. List all the numbers that divide it exactly (without leaving a remainder).
3. Step 3: Count the factors — Once you have listed all the factors, count how many there are.
4. Step 4: Classify based on the count — If the number has exactly two factors (1 and itself), it's a prime number. If the number has more than two factors, it's a composite number.

Worked Examples: Classifying Numbers

• Example 1: Classify the number 29 as even/odd and prime/composite. Even/Odd: Step 1: Look at the last digit of 29, which is 9. Step 2: Since 9 is an odd digit (1, 3, 5, 7, 9), 29 is an odd number. Prime/Composite: Step 1: Find factors of 29. Can 29 be divided by 2? No. By 3? No. By 5? No. By 7? No. The only numbers that divide 29 exactly are 1 and 29. Step 2: Count the factors. There are exactly two factors (1 and 29). * Step 3: Since it has exactly two factors, 29 is a prime number. Conclusion: 29 is an odd and prime number.
• Example 2: Classify the number 48 as even/odd and prime/composite. Even/Odd: Step 1: Look at the last digit of 48, which is 8. Step 2: Since 8 is an even digit (0, 2, 4, 6, 8), 48 is an even number. Prime/Composite: Step 1: Find factors of 48. We can divide 48 by 1 (48), by 2 (24), by 3 (16), by 4 (12), by 6 (8). The factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Step 2: Count the factors. There are many factors (more than two). * Step 3: Since it has more than two factors, 48 is a composite number. Conclusion: 48 is an even and composite number.
• Example 3: Identify a number that is both even and prime. Step 1: Recall the definition of an even number: divisible by 2. Step 2: Recall the definition of a prime number: has exactly two factors (1 and itself). Step 3: Start checking even numbers: 2, 4, 6, 8... Step 4: For 2: Factors are 1 and 2. It has exactly two factors. So, 2 is prime. * Step 5: For 4: Factors are 1, 2, 4. It has more than two factors. So, 4 is composite. Conclusion: The only number that is both even and prime is 2.

Important Points to Remember

• Number 1 is unique: It is neither prime nor composite. It has only one factor.
• Smallest Prime Number: The smallest prime number is 2. It is also the only even prime number.
• Smallest Composite Number: The smallest composite number is 4.
• Every even number (except 2) is composite: If an even number is greater than 2, it will always be divisible by 1, 2, and itself, hence having more than two factors.
• Prime numbers have a specific factor count: Always exactly two factors – 1 and the number itself.
• Composite numbers have diverse factor counts: Always more than two factors.

Practice Questions with Solutions

• Q: State whether the following statements are True or False: (a) The sum of three odd numbers is even. (b) The product of three odd numbers is odd. (c) The sum of an even number and an odd number is even. A: Step 1: Analyze statement (a). Let's take three odd numbers, e.g., 1, 3, 5. Their sum is 1 + 3 + 5 = 9. 9 is an odd number. So, the statement is False. Step 2: Analyze statement (b). Let's take three odd numbers, e.g., 1, 3, 5. Their product is 1 x 3 x 5 = 15. 15 is an odd number. This holds true for any three odd numbers (Odd x Odd = Odd; Odd x Odd = Odd). So, the statement is True. Step 3: Analyze statement (c). Let's take an even number (e.g., 2) and an odd number (e.g., 3). Their sum is 2 + 3 = 5. 5 is an odd number. So, the statement is False. Final answer: (a) False, (b) True, (c) False
• Q: What is the greatest prime number between 1 and 10? A: Step 1: List all numbers between 1 and 10: 2, 3, 4, 5, 6, 7, 8, 9. Step 2: Identify prime numbers from this list: - 2 (factors: 1, 2) - Prime - 3 (factors: 1, 3) - Prime - 4 (factors: 1, 2, 4) - Composite - 5 (factors: 1, 5) - Prime - 6 (factors: 1, 2, 3, 6) - Composite - 7 (factors: 1, 7) - Prime - 8 (factors: 1, 2, 4, 8) - Composite - 9 (factors: 1, 3, 9) - Composite Step 3: The prime numbers are 2, 3, 5, 7. The greatest among these is 7. Final answer: 7
• Q: Express the following as the sum of two odd primes: (a) 21 (b) 31 A: Step 1: For (a) 21. We need two odd prime numbers whose sum is 21. Let's try combining odd primes: - Start with smaller odd primes: 3, 5, 7, 11, 13, 17, 19... - Try 3 + ? = 21 -> ? = 18 (18 is not prime) - Try 5 + ? = 21 -> ? = 16 (16 is not prime) - Try 7 + ? = 21 -> ? = 14 (14 is not prime) - Try 11 + ? = 21 -> ? = 10 (10 is not prime) - Try 13 + ? = 21 -> ? = 8 (8 is not prime) - Try 17 + ? = 21 -> ? = 4 (4 is not prime) - Try 19 + ? = 21 -> ? = 2 (2 is prime, but it's even. The question asks for two odd primes). This means there might be a typo in the question or it's a tricky one. Let's recheck if 21 can be represented as a sum of two odd primes. Ah, this is a common trick. Let's consider Goldbach's Conjecture for even numbers, but this is an odd number. An odd number can be written as a sum of two primes only if one of them is 2. Since 2 is even, and the problem asks for two odd primes, it's impossible to express an odd number as the sum of two odd primes, because Odd + Odd = Even. Let's assume the question meant 'sum of an odd prime and an even prime' OR 'sum of two primes'. For Class 6, this question might be flawed. Let's re-interpret assuming the question meant to express an even number as sum of two odd primes. If not, the answer for (a) is 'Not possible as an odd number cannot be the sum of two odd numbers'. Self-correction: NCERT Ex 3.2 Q5 asks to express numbers as sum of two odd primes. It gives even numbers. So 21 and 31 are typos, they should be even numbers. Let's assume the question meant 20 and 30. Let's change (a) to 20 and (b) to 30 for validity in Class 6 context. (a) 20: Odd primes: 3, 5, 7, 11, 13, 17, 19. - 3 + 17 = 20. Both 3 and 17 are odd primes. (Correct) (b) 30: Odd primes: 3, 5, 7, 11, 13, 17, 19, 23, 29... - 7 + 23 = 30. Both 7 and 23 are odd primes. - 11 + 19 = 30. Both 11 and 19 are odd primes. - 13 + 17 = 30. Both 13 and 17 are odd primes. Final answer: (a) 20 = 3 + 17 (or 7 + 13), (b) 30 = 7 + 23 (or 11 + 19 or 13 + 17)
• Q: Write down separately the prime and composite numbers less than 20. A: Step 1: List numbers less than 20 (excluding 1 as it's neither): 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. Step 2: Go through the list and identify factors for each number. - Prime numbers (exactly two factors: 1 and itself): - 2 (factors: 1, 2) - 3 (factors: 1, 3) - 5 (factors: 1, 5) - 7 (factors: 1, 7) - 11 (factors: 1, 11) - 13 (factors: 1, 13) - 17 (factors: 1, 17) - 19 (factors: 1, 19) - Composite numbers (more than two factors): - 4 (factors: 1, 2, 4) - 6 (factors: 1, 2, 3, 6) - 8 (factors: 1, 2, 4, 8) - 9 (factors: 1, 3, 9) - 10 (factors: 1, 2, 5, 10) - 12 (factors: 1, 2, 3, 4, 6, 12) - 14 (factors: 1, 2, 7, 14) - 15 (factors: 1, 3, 5, 15) - 16 (factors: 1, 2, 4, 8, 16) - 18 (factors: 1, 2, 3, 6, 9, 18) Final answer: Prime numbers less than 20: 2, 3, 5, 7, 11, 13, 17, 19. Composite numbers less than 20: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18.

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