CBSE Class 6 Maths: Playing with Numbers Ex 3.1 - Factors and Multiples

Understanding Factors and Multiples

In our daily lives, we often distribute things or look for patterns. For example, if you have 12 candies and want to share them equally among your friends, you need to know how many friends can get an equal share. This is where factors come in! A factor of a number is a number that divides it exactly, leaving no remainder. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12, because 12 can be divided perfectly by each of these numbers.

On the other hand, multiples are like skip-counting. If you count by 5s, you get 5, 10, 15, 20, and so on. These numbers are all multiples of 5. A multiple of a number is what you get when you multiply that number by any whole number (like 1, 2, 3, ...). Understanding these two concepts is key to solving many problems in mathematics and seeing the structure in numbers.

Key Definitions

Factor

A factor of a number is an exact divisor of that number. When you divide a number by its factor, the remainder is always zero. For example, 3 is a factor of 15 because 15 ÷ 3 = 5 with no remainder.

Multiple

A multiple of a number is the product obtained when that number is multiplied by any whole number (1, 2, 3, 4, ...). For example, 20 is a multiple of 4 because 4 × 5 = 20.

How to Find Factors and Multiples

1. Finding Factors of a Number — To find all factors of a given number, you can follow these steps: 1. Start with 1. Every number has 1 as a factor. 2. Check divisibility by consecutive numbers (2, 3, 4, ...) up to the square root of the number. If a number divides it evenly, both the divisor and the quotient are factors. 3. Stop checking when the divisor becomes greater than the square root of the number or you start finding factors you've already identified as quotients. 4. List all unique factors in ascending order.
2. Finding Multiples of a Number — To find the multiples of a given number, simply multiply the number by 1, 2, 3, 4, and so on, for as many multiples as you need. 1. Multiply the number by 1 to get the first multiple. 2. Multiply the number by 2 to get the second multiple. 3. Continue this process for the desired number of multiples. Remember, multiples are infinite!

Worked Examples

• Example 1: Find all factors of 18. Step 1: Start checking numbers from 1. 1 × 18 = 18 (So, 1 and 18 are factors) Step 2: Check for 2. 2 × 9 = 18 (So, 2 and 9 are factors) Step 3: Check for 3. 3 × 6 = 18 (So, 3 and 6 are factors) Step 4: Check for 4. 18 ÷ 4 is not exact. Step 5: Check for 5. 18 ÷ 5 is not exact. Step 6: Check for 6. We already found 6. We can stop here. Final Answer: The factors of 18 are 1, 2, 3, 6, 9, 18.
• Example 2: Write the first 5 multiples of 8. Step 1: Multiply 8 by 1. 8 × 1 = 8 Step 2: Multiply 8 by 2. 8 × 2 = 16 Step 3: Multiply 8 by 3. 8 × 3 = 24 Step 4: Multiply 8 by 4. 8 × 4 = 32 Step 5: Multiply 8 by 5. 8 × 5 = 40 Final Answer: The first 5 multiples of 8 are 8, 16, 24, 32, 40.
• Example 3: Is 5 a factor of 45? Is 45 a multiple of 5? Step 1: To check if 5 is a factor of 45, divide 45 by 5. 45 ÷ 5 = 9. Since the remainder is 0, 5 is an exact divisor. Step 2: To check if 45 is a multiple of 5, see if 45 can be obtained by multiplying 5 by a whole number. 5 × 9 = 45. Final Answer: Yes, 5 is a factor of 45. And yes, 45 is a multiple of 5.

Important Things to Remember

• 1 is a factor of every number. No matter what number you pick, 1 will always divide it exactly.
• Every number is a factor of itself. For example, 7 is a factor of 7 (7 ÷ 7 = 1).
• Every number is a multiple of itself. For example, 7 is a multiple of 7 (7 × 1 = 7).
• The factors of a number are finite. There's a limited list of factors for any given number.
• The multiples of a number are infinite. You can keep multiplying a number by 1, 2, 3, 4... forever!
• The smallest factor of a number is 1.
• The largest factor of a number is the number itself.
• The smallest multiple of a number is the number itself.

Practice Questions with Solutions

• Q: Write all factors of 28. A: Step 1: Start checking divisors from 1. 1 × 28 = 28 2 × 14 = 28 4 × 7 = 28 Step 2: Check numbers between 4 and 7 (i.e., 5, 6). Neither 5 nor 6 divides 28 exactly. The next factor after 4 would be 7, which we already found. Final answer: The factors of 28 are 1, 2, 4, 7, 14, 28.
• Q: Write the first 4 multiples of 12. A: Step 1: To find the first multiple, multiply 12 by 1. 12 × 1 = 12 Step 2: To find the second multiple, multiply 12 by 2. 12 × 2 = 24 Step 3: To find the third multiple, multiply 12 by 3. 12 × 3 = 36 Step 4: To find the fourth multiple, multiply 12 by 4. 12 × 4 = 48 Final answer: The first 4 multiples of 12 are 12, 24, 36, 48.
• Q: Is 9 a factor of 99? Justify your answer. A: Step 1: To check if 9 is a factor of 99, we need to see if 9 divides 99 exactly, meaning with no remainder. Step 2: Perform the division: 99 ÷ 9 = 11. Step 3: Since the remainder is 0, 9 divides 99 exactly. Final answer: Yes, 9 is a factor of 99 because 99 divided by 9 gives a whole number (11) with no remainder.
• Q: Is 60 a multiple of 7? Justify your answer. A: Step 1: To check if 60 is a multiple of 7, we need to see if 60 can be obtained by multiplying 7 by a whole number. Step 2: Divide 60 by 7: 60 ÷ 7 = 8 with a remainder of 4. (7 × 8 = 56, 7 × 9 = 63). Step 3: Since 60 does not divide by 7 exactly (it leaves a remainder), it is not a multiple of 7. Final answer: No, 60 is not a multiple of 7 because when 60 is divided by 7, there is a remainder (4), meaning 7 does not multiply by a whole number to give 60.

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