Playing with Numbers Exercise 3.6 Class 6 NCERT Solutions

Understanding the Highest Common Factor (HCF)

The Highest Common Factor (HCF), also sometimes called the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. To understand HCF, let's first quickly recall what factors are. A factor of a number is a number that divides it exactly. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18.

Now, let's look for the common factors between 12 and 18. These are the numbers that appear in both lists: 1, 2, 3, and 6. Among these common factors, the highest one is 6. So, the HCF of 12 and 18 is 6.

HCF is incredibly useful! Imagine you have 12 apples and 18 oranges, and you want to pack them into identical bags such that each bag has the same number of apples and the same number of oranges, with no fruit left over. The maximum number of such bags you can make is 6 (the HCF!). Each bag would have 2 apples (12 ÷ 6) and 3 oranges (18 ÷ 6). This concept is fundamental to many mathematical operations and problem-solving scenarios.

Finding HCF using Prime Factorization Method

1. Step 1: Find Prime Factorization of Each Number — Break down each given number into its prime factors. Remember, prime factors are prime numbers (like 2, 3, 5, 7, 11...) that multiply together to give the original number. For example, for 24, its prime factorization is 2 x 2 x 2 x 3, or 2³ × 3¹.
2. Step 2: Identify Common Prime Factors — Look at the prime factorizations of all the numbers. Identify all the prime factors that are common to ALL of them. If a prime factor is present in one number's factorization but not another's, it is NOT a common prime factor.
3. Step 3: Take the Lowest Power of Each Common Prime Factor — For each common prime factor you identified in Step 2, choose the smallest power (or the lowest number of times it appears) across all the prime factorizations. For instance, if '2' appears as 2³ in one number and 2² in another, you would take 2².
4. Step 4: Multiply These Selected Prime Factors — Multiply all the prime factors you selected in Step 3 (each raised to its chosen lowest power). The product will be the HCF of the given numbers. If there are no common prime factors other than 1 (meaning the numbers are coprime), their HCF is 1.

Worked Examples: Calculating HCF

• Example 1: Find the HCF of 20 and 30. Step 1: Prime Factorization 20 = 2 × 2 × 5 = 2² × 5¹ 30 = 2 × 3 × 5 = 2¹ × 3¹ × 5¹ Step 2: Identify Common Prime Factors The common prime factors are 2 and 5. Step 3: Take the Lowest Power For prime factor 2: The lowest power is 2¹ (from 30). For prime factor 5: The lowest power is 5¹ (from both 20 and 30). Step 4: Multiply HCF = 2¹ × 5¹ = 2 × 5 = 10. Final Answer: The HCF of 20 and 30 is 10.
• Example 2: Find the HCF of 36, 48, and 60. Step 1: Prime Factorization 36 = 2 × 2 × 3 × 3 = 2² × 3² 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3¹ 60 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹ Step 2: Identify Common Prime Factors The common prime factors are 2 and 3. (5 is not common as it's not in 36 or 48). Step 3: Take the Lowest Power For prime factor 2: The lowest power is 2² (from 36 and 60). For prime factor 3: The lowest power is 3¹ (from 48 and 60). Step 4: Multiply HCF = 2² × 3¹ = 4 × 3 = 12. Final Answer: The HCF of 36, 48, and 60 is 12.

Exam Tip: Avoiding Common HCF Mistakes

When calculating HCF, especially with prime factorization, students often make a few common mistakes. One frequent error is confusing HCF with LCM (Least Common Multiple) – remember, HCF takes the lowest powers of common prime factors, while LCM takes the highest powers of all prime factors. Another mistake is forgetting to find the prime factors accurately; always double-check your prime factorization. Also, ensure you only select prime factors that are common to all the numbers you are finding the HCF for. If a number does not have a particular prime factor, then it cannot be a common factor for the group.

Practice Questions with Solutions

• Q: Find the HCF of 18 and 48. A: Step 1: Prime factorize each number. 18 = 2 × 3 × 3 = 2¹ × 3² 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3¹ Step 2: Identify common prime factors: 2 and 3. Step 3: Take the lowest power for each common prime factor. For 2: lowest power is 2¹. For 3: lowest power is 3¹. Step 4: Multiply the selected factors. HCF = 2¹ × 3¹ = 2 × 3 = 6. Final answer: The HCF of 18 and 48 is 6.
• Q: What is the HCF of 27, 36, and 63? A: Step 1: Prime factorize each number. 27 = 3 × 3 × 3 = 3³ 36 = 2 × 2 × 3 × 3 = 2² × 3² 63 = 3 × 3 × 7 = 3² × 7¹ Step 2: Identify common prime factors: Only 3 is common to all three numbers. Step 3: Take the lowest power for the common prime factor. For 3: lowest power is 3² (from 36 and 63). Step 4: Multiply the selected factors. HCF = 3² = 9. Final answer: The HCF of 27, 36, and 63 is 9.
• Q: Two wires are 75 cm and 120 cm long. What is the maximum length of a piece that can be cut from both wires exactly, without any leftover? A: Step 1: This is an HCF problem, as we need the 'maximum' length that divides 'exactly' into both numbers. Prime factorize 75 and 120. 75 = 3 × 5 × 5 = 3¹ × 5² 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3¹ × 5¹ Step 2: Identify common prime factors: 3 and 5. Step 3: Take the lowest power for each common prime factor. For 3: lowest power is 3¹. For 5: lowest power is 5¹. Step 4: Multiply the selected factors. HCF = 3¹ × 5¹ = 3 × 5 = 15. Final answer: The maximum length of a piece is 15 cm.
• Q: Find the HCF of 42 and 98. A: Step 1: Prime factorize each number. 42 = 2 × 3 × 7 = 2¹ × 3¹ × 7¹ 98 = 2 × 7 × 7 = 2¹ × 7² Step 2: Identify common prime factors: 2 and 7. Step 3: Take the lowest power for each common prime factor. For 2: lowest power is 2¹. For 7: lowest power is 7¹. Step 4: Multiply the selected factors. HCF = 2¹ × 7¹ = 2 × 7 = 14. Final answer: The HCF of 42 and 98 is 14.

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