How to Solve: Number System Basics (LCM, HCF, Divisibility)

How to Solve: Number System Basics (LCM, HCF, Divisibility)

For SSC / Bank / Railway Exams

Introduction

"Master LCM, HCF, and divisibility rules, and you’ll solve 5-7 questions in every SSC/Bank/Railway exam—worth 10-15 marks—faster than your competitors. These concepts are the backbone of number systems, and examiners test them in every shift. Let’s break them down so you never lose a mark here again."

What You Need To Know First

1. Prime Numbers: Numbers >1 with no divisors other than 1 and itself (e.g., 2, 3, 5, 7).
2. Factors & Multiples: If a divides b exactly, a is a factor of b, and b is a multiple of a.
3. Division Basics: Dividend = (Divisor × Quotient) + Remainder.

Key Vocabulary

Formulas To Know

1. LCM of Two Numbers
   Formula: LCM(a, b) = (a × b) ÷ HCF(a, b)
   Variables:
2. a, b = two numbers.

3. MEMORISE THIS (not given in exams).

4. HCF of Two Numbers (Prime Factorisation Method)

5. Break both numbers into prime factors.
6. Multiply the common prime factors with the lowest powers.

7. MEMORISE THIS.

8. Divisibility Rules (Shortcuts)
   | Divisor | Rule | Example |
   |-------------|--------------------------------------------------------------------------|---------------------------------|
   | 2 | Last digit is even (0, 2, 4, 6, 8). | 246 → divisible by 2. |
   | 3 | Sum of digits is divisible by 3. | 123 → 1+2+3=6 → divisible. |
   | 4 | Last two digits form a number divisible by 4. | 1324 → 24 ÷ 4 = 6 → divisible. |
   | 5 | Last digit is 0 or 5. | 125 → divisible. |
   | 6 | Divisible by both 2 and 3. | 132 → even + sum=6 → divisible. |
   | 8 | Last three digits form a number divisible by 8. | 1048 → 048 ÷ 8 = 6 → divisible. |
   | 9 | Sum of digits is divisible by 9. | 729 → 7+2+9=18 → divisible. |
   | 10 | Last digit is 0. | 150 → divisible. |

9. MEMORISE THESE (exams test these frequently).

10. Product of Two Numbers
    Formula: a × b = LCM(a, b) × HCF(a, b)

11. MEMORISE THIS (useful for solving missing values).

Step-by-Step Method

How to Find LCM (Prime Factorisation Method)

1. Break both numbers into prime factors.
2. Example: Find LCM of 12 and 18.
   • 12 = 2² × 3¹
   • 18 = 2¹ × 3²
3. Take the highest power of each prime factor.
4. 2² (from 12) and 3² (from 18).
5. Multiply them together.
6. LCM = 2² × 3² = 4 × 9 = 36.

How to Find HCF (Prime Factorisation Method)

1. Break both numbers into prime factors.
2. Example: Find HCF of 12 and 18.
   • 12 = 2² × 3¹
   • 18 = 2¹ × 3²
3. Take the lowest power of common prime factors.
4. Common primes: 2 and 3.
5. Lowest powers: 2¹ and 3¹.
6. Multiply them together.
7. HCF = 2¹ × 3¹ = 2 × 3 = 6.

How to Check Divisibility (Quick Rules)

1. Identify the divisor (e.g., 3, 4, 5, etc.).
2. Apply the rule (see table above).
3. Verify by dividing if unsure.

Worked Examples

Example 1 – Basic LCM & HCF

Question: Find LCM and HCF of 24 and 36. Solution: 1. Prime factors:
- 24 = 2³ × 3¹
- 36 = 2² × 3² 2. LCM: Highest powers → 2³ × 3² = 8 × 9 = 72. 3. HCF: Lowest powers → 2² × 3¹ = 4 × 3 = 12. What we did and why: - Broke numbers into primes to compare factors systematically. - LCM needs the "biggest" factors; HCF needs the "smallest" common factors.

Example 2 – Medium (LCM with 3 Numbers)

Question: Find LCM of 4, 6, and 8. Solution: 1. Prime factors:
- 4 = 2²
- 6 = 2¹ × 3¹
- 8 = 2³ 2. Highest powers: 2³ × 3¹ = 8 × 3 = 24. What we did and why: - Extended the method to 3 numbers by including all primes. - Ensured no prime was missed (e.g., 3 from 6).

Example 3 – Exam-Style (Disguised HCF)

Question: The product of two numbers is 180. Their HCF is 6. Find their LCM. Solution: 1. Use the formula: a × b = LCM × HCF. 2. Plug in values: 180 = LCM × 6. 3. Solve: LCM = 180 ÷ 6 = 30. What we did and why: - Recognised the formula connects product, LCM, and HCF. - Avoided unnecessary prime factorisation by using the shortcut.

Common Mistakes

Exam Traps

1-Minute Recap

"Listen up—this is your last-minute checklist for LCM, HCF, and divisibility: 1. LCM: Prime factors → highest powers → multiply. 2. HCF: Prime factors → lowest common powers → multiply. 3. Divisibility: Memorise the rules for 2, 3, 4, 5, 6, 8, 9, 10. 4. Formula: a × b = LCM × HCF. Use it to save time. 5. Traps: Watch for co-prime numbers, large digit sums, and fractions. Now go solve 5 questions in a row—you’ve got this!