Number System

Number System

Overview

Mastering HCF/LCM, divisibility rules, and fraction-decimal conversions will significantly speed up your calculations across the entire Numerical Ability section.

This topic rewards memorization of key rules and patterns rather than complex problem-solving. Students who internalize divisibility tests and know standard fraction-decimal equivalents can solve these questions in under 30 seconds, freeing up time for lengthier DI sets. The concepts here are straightforward but require consistent practice to achieve the speed needed for Prelims.

Key Concepts

• HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Also called GCD. Used when dividing things into equal groups or finding common measurements.
• LCM (Least Common Multiple): The smallest number that is divisible by two or more numbers. Used for problems involving repetition, cycles, or finding when events coincide.
• HCF × LCM = Product of two numbers: This relationship holds only for two numbers and is frequently tested.
• Co-prime numbers: Two numbers with HCF = 1. Example: 8 and 15 are co-prime.
• Divisibility rules: Shortcuts to check if a number is divisible by 2, 3, 4, 5, 6, 8, 9, 11 without actual division.
• Fractions to decimals: Conversion requires either direct division or memorization of standard equivalents (1/8 = 0.125, 1/6 = 0.1666...).
• Types of fractions: Proper (numerator < denominator), Improper (numerator ≥ denominator), Mixed (whole + proper fraction).
• Recurring vs. terminating decimals: Decimals terminate when the denominator has only 2 and 5 as prime factors; otherwise, they recur.

Formulas / Key Facts

HCF and LCM:

• HCF × LCM = a × b (for two numbers a and b)
• HCF of fractions = HCF of numerators ÷ LCM of denominators
• LCM of fractions = LCM of numerators ÷ HCF of denominators

Divisibility Rules:

Standard Fraction-Decimal Conversions (memorize these):

• 1/2 = 0.5
• 1/3 = 0.333...
• 1/4 = 0.25
• 1/5 = 0.2
• 1/6 = 0.1666...
• 1/7 = 0.142857 (repeating)
• 1/8 = 0.125
• 1/9 = 0.111...
• 1/11 = 0.0909...
• 1/12 = 0.0833...

Worked Examples

Example 1: Finding HCF and LCM

Find HCF and LCM of 12 and 18.

Step 1: Prime factorization

• 12 = 2² × 3
• 18 = 2 × 3²

Step 2: HCF = Product of lowest powers of common primes

• HCF = 2¹ × 3¹ = 6

Step 3: LCM = Product of highest powers of all primes

• LCM = 2² × 3² = 36

Verification: HCF × LCM = 6 × 36 = 216 = 12 × 18 ✓

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Example 2: Application of HCF-LCM relationship

The HCF of two numbers is 8 and their LCM is 96. If one number is 24, find the other.

Using the formula: HCF × LCM = First number × Second number

• 8 × 96 = 24 × Second number
• 768 = 24 × Second number
• Second number = 768 ÷ 24 = 32

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Example 3: Divisibility check

Is 3,528 divisible by 8?

Rule for 8: Check last three digits

• Last three digits = 528
• 528 ÷ 8 = 66

Yes, 3,528 is divisible by 8.

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Example 4: HCF of fractions

Find HCF of 2/3, 4/5, and 6/7.

HCF of fractions = HCF of numerators ÷ LCM of denominators

• HCF of 2, 4, 6 = 2
• LCM of 3, 5, 7 = 105

HCF = 2/105

Common Mistakes

• Confusing HCF and LCM formulas for fractions → Remember: HCF uses HCF on top and LCM below; LCM uses LCM on top and HCF below. Think "opposite operations."
• Applying HCF × LCM = Product formula to three or more numbers → This formula works ONLY for exactly two numbers. For three numbers, you need a different approach.
• Forgetting to check both conditions for divisibility by 6 → A number must be divisible by BOTH 2 AND 3 to be divisible by 6. Checking only one condition leads to wrong answers.
• Mixing up divisibility rules for 4 and 8 → For 4: check last TWO digits. For 8: check last THREE digits. Associate 4 with 2 digits, 8 with 3 digits.
• Converting recurring decimals incorrectly → 0.333... equals 1/3, not 0.33. When comparing, either convert both to fractions or use sufficient decimal places.

Quick Reference

• HCF × LCM = Product (only for 2 numbers)
• Divisible by 3: digit sum divisible by 3
• Divisible by 4: last 2 digits divisible by 4
• Divisible by 8: last 3 digits divisible by 8
• Divisible by 11: (sum of odd-position digits) − (sum of even-position digits) divisible by 11
• 1/8 = 0.125, 1/6 = 0.1666..., 1/7 = 0.142857