How to Solve: HCF Problems

How to Solve: HCF Problems

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Imagine you’re packing identical gift bags for a party—how many bags can you make without leftovers? That’s HCF in real life, and it’s a guaranteed 2-3 marks on your exam. Let’s master it in 10 minutes."

What You Need To Know First

Before diving into HCF, ensure you understand: 1. Factors: Numbers that divide another number exactly (e.g., factors of 12: 1, 2, 3, 4, 6, 12). 2. Prime Numbers: Numbers with only two factors (1 and itself, e.g., 2, 3, 5, 7). 3. Division Basics: How to divide numbers and check for remainders.

Key Vocabulary

Formulas To Know

1. Prime Factorisation Method

Formula: Break numbers into prime factors, then multiply the common primes with the lowest powers. Variables: - Let the numbers be A and B. - Find prime factors of A and B. - Identify common primes. - Multiply the lowest power of each common prime.

MEMORISE THIS: This is the most reliable method for exams.

Example: Find HCF of 24 and 36. - 24 = 2³ × 3¹ - 36 = 2² × 3² - Common primes: 2 and 3. - Lowest powers: 2² × 3¹ = 12 (HCF).

2. Division (Euclidean) Method

Formula: Repeatedly divide the larger number by the smaller number until the remainder is 0. The last non-zero remainder is the HCF. Steps: 1. Divide A by B, find remainder R. 2. Replace A with B, and B with R. 3. Repeat until R = 0. The last B is the HCF.

MEMORISE THIS: Best for large numbers (e.g., 126, 180).

Example: Find HCF of 48 and 18. - 48 ÷ 18 = 2 remainder 12. - 18 ÷ 12 = 1 remainder 6. - 12 ÷ 6 = 2 remainder 0. - HCF = 6.

3. Listing Factors Method

Formula: List all factors of each number, then pick the largest common one. When to use: Only for small numbers (e.g., ≤ 50).

Example: Find HCF of 15 and 20. - Factors of 15: 1, 3, 5, 15. - Factors of 20: 1, 2, 4, 5, 10, 20. - Common factors: 1, 5. - HCF = 5.

Step-by-Step Method

(Use this for every HCF problem on your exam.)

Step 1: Read the Problem Carefully

• Underline the numbers involved.
• Check if it’s a word problem (e.g., "identical groups," "maximum size").

Step 2: Choose the Best Method

• Small numbers (≤ 50): Listing factors.
• Medium numbers (50–200): Prime factorisation.
• Large numbers (> 200): Division method.

Step 3: Apply the Method

• Prime Factorisation:
• Break each number into primes.
• Circle common primes.
• Multiply the lowest power of each common prime.
• Division Method:
• Divide larger by smaller, find remainder.
• Replace numbers, repeat until remainder = 0.
• Last non-zero remainder = HCF.
• Listing Factors:
• List all factors of each number.
• Identify common factors.
• Pick the largest one.

Step 4: Verify Your Answer

• Check if the HCF divides all original numbers exactly.
• If not, recheck your steps.

Step 5: Answer the Question

• If it’s a word problem, write a full sentence (e.g., "The maximum number of identical bags is 6.").

Worked Examples

Example 1 – Basic (Prime Factorisation)

Problem: Find the HCF of 36 and 48.

Step-by-Step Solution: 1. Prime factors of 36:
36 = 2 × 2 × 3 × 3 = 2² × 3². 2. Prime factors of 48:
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3¹. 3. Common primes: 2 and 3. 4. Lowest powers: 2² × 3¹ = 4 × 3 = 12. 5. Verify: 36 ÷ 12 = 3, 48 ÷ 12 = 4 (no remainders).

Answer: 12.

What we did and why: - Used prime factorisation because the numbers are medium-sized. - Multiplied the lowest powers of common primes to ensure the HCF is the largest possible.

Example 2 – Medium (Division Method)

Problem: Find the HCF of 126 and 180.

Step-by-Step Solution: 1. Divide 180 by 126:
180 ÷ 126 = 1 remainder 54. 2. Divide 126 by 54:
126 ÷ 54 = 2 remainder 18. 3. Divide 54 by 18:
54 ÷ 18 = 3 remainder 0. 4. Last non-zero remainder: 18.

Answer: 18.

What we did and why: - Used the division method because the numbers are large. - Repeated division until remainder = 0 to find the HCF efficiently.

Example 3 – Exam Style (Word Problem)

Problem: A teacher has 40 pencils and 24 erasers. She wants to make identical gift bags with no leftovers. What is the maximum number of bags she can make?

Step-by-Step Solution: 1. Understand the problem: We need the largest number that divides 40 and 24 exactly. 2. Find HCF of 40 and 24:
- Prime factors of 40: 2 × 2 × 2 × 5 = 2³ × 5¹.
- Prime factors of 24: 2 × 2 × 2 × 3 = 2³ × 3¹.
- Common prime: 2.
- Lowest power: 2³ = 8. 3. Verify: 40 ÷ 8 = 5, 24 ÷ 8 = 3 (no remainders). 4. Answer the question: The maximum number of bags is 8.

What we did and why: - Recognised that "identical bags with no leftovers" means HCF. - Used prime factorisation to find the largest common divisor. - Wrote a full-sentence answer to match exam expectations.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, listen up—this is your 60-second HCF survival guide. First, HCF is just the biggest number that divides two (or more) numbers exactly. Got it? Good.

Three methods—pick the right one: 1. Small numbers? List all factors, pick the biggest common one. 2. Medium numbers? Prime factorisation—break them into primes, multiply the lowest power of common primes. 3. Big numbers? Division method—divide, replace, repeat until remainder is 0.

Word problems? Look for keywords: "maximum," "identical," "no leftovers." That’s HCF.

Common traps? Don’t multiply all common primes—only the lowest powers. And if there are three numbers, do two first, then the third.

Last tip: Always verify your answer divides all numbers exactly. No remainders? You’re golden.

Now go crush that exam. You’ve got this!