How to Solve: Number System Basics

How to Solve: Number System Basics

Introduction

"If you can’t tell the difference between a prime number and a composite number, you’ll lose marks on every arithmetic question—from fractions to algebra. Let’s fix that in 10 minutes."

What You Need To Know First

Before diving into number systems, ensure you understand: 1. Place Value – How digits represent values (e.g., in 345, 4 is in the "tens" place). 2. Basic Operations – Addition, subtraction, multiplication, and division of whole numbers. 3. Factors & Multiples – What it means for a number to be a factor or multiple of another (e.g., 3 is a factor of 12).

Key Vocabulary

Formulas To Know

Step-by-Step Method

How to Identify Prime vs. Composite Numbers

1. Check if the number is 1 → 1 is neither prime nor composite.
2. Check if the number is 2 → 2 is the only even prime number.
3. For numbers >2:
4. Try dividing by 2 → If it divides evenly, it’s composite.
5. Try dividing by 3 → If it divides evenly, it’s composite.
6. Continue up to √n (square root of the number). If no divisors, it’s prime.

How to Find HCF (Highest Common Factor)

1. List all factors of each number.
2. Identify common factors.
3. Pick the largest one.

OR (Faster Method – Prime Factorisation) 1. Break each number into prime factors. 2. Circle the common primes (lowest power). 3. Multiply them together.

How to Find LCM (Lowest Common Multiple)

1. List multiples of each number until you find a common one.
2. Pick the smallest common multiple.

OR (Faster Method – Prime Factorisation) 1. Break each number into prime factors. 2. Take the highest power of each prime. 3. Multiply them together.

Worked Examples

Example 1 – Basic: Is 17 a Prime Number?

Step 1: 17 > 1 → Not 1. Step 2: 17 is odd → Not divisible by 2. Step 3: Sum of digits (1 + 7 = 8) → Not divisible by 3. Step 4: Last digit is 7 → Not divisible by 5. Step 5: Check divisibility up to √17 (~4.1) → Only 1 and 17 divide it. Answer: 17 is prime.

What we did and why: - We checked divisibility by small primes (2, 3, 5) first because they rule out most composites quickly. - We stopped at √17 because if a number has no factors ≤ its square root, it’s prime.

Example 2 – Medium: Find HCF and LCM of 12 and 18

Step 1: Prime Factorisation - 12 = 2 × 2 × 3 = 2² × 3¹ - 18 = 2 × 3 × 3 = 2¹ × 3²

Step 2: HCF (Lowest Power of Common Primes) - Common primes: 2 and 3 - Lowest power of 2: 2¹ - Lowest power of 3: 3¹ - HCF = 2¹ × 3¹ = 6

Step 3: LCM (Highest Power of All Primes) - All primes: 2 and 3 - Highest power of 2: 2² - Highest power of 3: 3² - LCM = 2² × 3² = 36

Answer: HCF = 6, LCM = 36

What we did and why: - We used prime factorisation because it’s faster than listing all factors/multiples. - For HCF, we took the lowest power of common primes. - For LCM, we took the highest power of all primes.

Example 3 – Exam Style: Three bells ring at intervals of 6, 8, and 12 minutes. If they ring together at 9 AM, when will they next ring together?

Step 1: Understand the problem → We need the LCM of 6, 8, and 12. Step 2: Prime Factorisation - 6 = 2 × 3 - 8 = 2 × 2 × 2 = 2³ - 12 = 2 × 2 × 3 = 2² × 3¹

Step 3: Find LCM (Highest Power of All Primes) - Highest power of 2: 2³ - Highest power of 3: 3¹ - LCM = 2³ × 3¹ = 24

Step 4: Add to start time → 9 AM + 24 minutes = 9:24 AM

Answer: The bells will next ring together at 9:24 AM.

What we did and why: - We recognised that the problem was about LCM (next common multiple). - We used prime factorisation for speed. - We added the LCM to the start time to get the final answer.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in. Number systems are the foundation of math—mess this up, and fractions, algebra, even geometry will trip you up. Here’s the cheat sheet:

1. Primes have only two factors: 1 and themselves. 2 is the only even prime. 1 is not prime.
2. HCF is the biggest number that divides two numbers. Use prime factors and take the lowest power of common primes.
3. LCM is the smallest number both numbers divide into. Use prime factors and take the highest power of all primes.
4. Divisibility rules save time—memorise them for 2, 3, 5, and 10.
5. Exam traps? Watch for 1, 0, and word problems disguising LCM/HCF.

Tonight, grab a past paper and do three number system questions. Check your answers, fix mistakes, and you’ll walk into that exam confident. You’ve got this!

This guide ensures full marks on number system questions—whether you’re a student or a teacher prepping a lesson. Now go ace that exam! ?