Playing with Numbers Ex 3.4 NCERT Solutions for Class 6 Maths

Hello! Welcome to this lesson on Exercise 3.4 from the chapter 'Playing with Numbers'. So far, you've learned about factors and multiples. Now, we're going to take it a step further and find what's 'common' between different numbers. Think of it like finding friends who like the same cartoon as you! In this section, we'll master finding common factors and common multiples. We'll also learn about a special type of number pair called co-prime numbers. Understanding these ideas is super important because they are the building blocks for bigger topics like HCF (Highest Common Factor) and LCM (Lowest Common Multiple) which you'll use throughout your maths journey. Let's start playing with numbers together!

How to Find Common Factors

  1. Step 1: List all factors of the first number. — Remember, factors are numbers that divide a given number completely without leaving any remainder. For example, to find the factors of 12, think of all the pairs of numbers that multiply to give 12: 1 x 12, 2 x 6, 3 x 4. So the factors are 1, 2, 3, 4, 6, and 12.
  2. Step 2: List all factors of the second number. — Do the same thing for the second number. For example, if the second number is 18, the factors are: 1 x 18, 2 x 9, 3 x 6. So the factors are 1, 2, 3, 6, 9, and 18.
  3. Step 3: Identify the numbers that appear in both lists. — Look at both lists of factors you've made. Circle or write down the numbers that are present in both. For our example of 12 and 18, the lists are (1, 2, 3, 4, 6, 12) and (1, 2, 3, 6, 9, 18). The numbers that are in both lists are 1, 2, 3, and 6. These are the common factors!

Worked Examples: Finding Common Multiples

  • Example: Find the first three common multiples of 4 and 6. Step 1: Write the multiples of the first number (4). Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ... (This list goes on forever!) Step 2: Write the multiples of the second number (6). Multiples of 6 are: 6, 12, 18, 24, 30, 36, ... Step 3: Find the numbers common to both lists. Looking at both lists, the first number to appear in both is 12. The next number to appear in both is 24. The next one is 36. Final Answer: The first three common multiples of 4 and 6 are 12, 24, and 36.
  • Example: Find the first two common multiples of 5 and 10. Step 1: Write the multiples of 5. Multiples of 5 are: 5, 10, 15, 20, 25, 30, ... Step 2: Write the multiples of 10. Multiples of 10 are: 10, 20, 30, 40, ... Step 3: Identify the first two common numbers. The first common multiple is 10. The second common multiple is 20. Final Answer: The first two common multiples of 5 and 10 are 10 and 20.

What Are Co-prime Numbers?

This is a very interesting idea in mathematics! Two numbers are called co-prime if their only common factor is the number 1. It's like two people who are very different and have only one tiny thing in common.

Let's take an example. Consider the numbers 8 and 9.

  • Factors of 8 are: 1, 2, 4, 8.
  • Factors of 9 are: 1, 3, 9.

If you look at both lists, the only number that appears in both is 1. So, 8 and 9 are co-prime.

A common mistake is to think that co-prime numbers must be prime numbers. This is not true! In our example, neither 8 nor 9 is a prime number, but they are co-prime to each other.

Here's another example: 15 and 16.

  • Factors of 15: 1, 3, 5, 15
  • Factors of 16: 1, 2, 4, 8, 16

The only common factor is 1. So, 15 and 16 are co-prime.

What about numbers that are NOT co-prime? Let's see 12 and 18.

  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 18: 1, 2, 3, 6, 9, 18

Their common factors are 1, 2, 3, and 6. Since they have common factors other than 1, they are not co-prime.

Key Points to Remember

  • The number 1 is a common factor for any pair of numbers you can think of.
  • Two consecutive numbers (like 7 and 8, or 20 and 21) are always co-prime.
  • Don't confuse co-prime numbers with prime numbers. Co-prime is a relationship between two numbers, while a prime number is a property of a single number.
  • When asked for common multiples, read the question carefully. It will usually ask for the 'first two' or 'first three' common multiples because the list of common multiples is infinite.

Practice Questions with Solutions

  • Q: Find all the common factors of 20 and 28. A: Step 1: List the factors of 20. 1 x 20 = 20 2 x 10 = 20 4 x 5 = 20 The factors of 20 are 1, 2, 4, 5, 10, 20. Step 2: List the factors of 28. 1 x 28 = 28 2 x 14 = 28 4 x 7 = 28 The factors of 28 are 1, 2, 4, 7, 14, 28. Step 3: Identify the common numbers in both lists. The numbers that appear in both lists are 1, 2, and 4. Final answer: The common factors of 20 and 28 are 1, 2, and 4.
  • Q: Find all the common factors of 15 and 25. A: Step 1: List the factors of 15. Factors of 15 are 1, 3, 5, 15. Step 2: List the factors of 25. Factors of 25 are 1, 5, 25. Step 3: Find the common factors from both lists. The common numbers are 1 and 5. Final answer: The common factors of 15 and 25 are 1 and 5.
  • Q: Find the first three common multiples of 3 and 4. A: Step 1: List the multiples of 3. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ... Step 2: List the multiples of 4. Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ... Step 3: Identify the first three numbers that appear in both lists. The first common multiple is 12. The second common multiple is 24. The third common multiple is 36. Final answer: The first three common multiples of 3 and 4 are 12, 24, and 36.
  • Q: Are the numbers 18 and 35 co-prime? Explain why. A: Step 1: Find the factors of 18. Factors of 18 are 1, 2, 3, 6, 9, 18. Step 2: Find the factors of 35. Factors of 35 are 1, 5, 7, 35. Step 3: Check the common factors. The only number that is in both lists of factors is 1. Step 4: Conclude based on the definition of co-prime numbers. Since the only common factor is 1, the numbers are co-prime. Final answer: Yes, 18 and 35 are co-prime because their only common factor is 1.

Frequently Asked Questions

What is the main difference between a factor and a multiple?

A factor is a number that divides another number exactly. For example, 3 is a factor of 12. A multiple is the result of multiplying a number by an integer. For example, 12 is a multiple of 3.

How are common factors related to the Highest Common Factor (HCF)?

Common factors are all the factors that two or more numbers share. The Highest Common Factor (HCF) is simply the largest number among those common factors. For example, the common factors of 12 and 18 are 1, 2, 3, 6, and the HCF is 6.

Are any two prime numbers always co-prime?

Yes! A prime number has only two factors: 1 and itself. So, if you take any two different prime numbers, like 7 and 11, their only common factor will always be 1. This makes them co-prime.