Number System
Study Notes for MPESB Group 1/2/3
Overview
The Number System forms the foundation of quantitative aptitude in MPESB exams. Questions from this topic appear directly (identifying number types, finding LCM/HCF) and indirectly (as building blocks for percentages, ratios, and simplification problems).
Mastery requires two things: first, crystal-clear classification of number types (whole, rational, irrational); second, fluency with LCM, HCF, and divisibility rules that enable quick mental calculations. Students who invest time here find the entire quantitative section easier because these concepts recur everywhere.
The scope is well-defined: know your number classifications, memorize divisibility tests up to 11, and practice LCM/HCF problems until the methods become automatic.
Key Concepts
- Natural Numbers (N): Counting numbers starting from 1. Set: {1, 2, 3, 4, ...}. Zero is NOT included.
- Whole Numbers (W): Natural numbers plus zero. Set: {0, 1, 2, 3, ...}. Every natural number is a whole number.
- Integers (Z): Whole numbers plus negatives. Set: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Includes positive, negative, and zero.
- Rational Numbers (Q): Any number expressible as p/q where p and q are integers and q ≠ 0. Includes terminating decimals (0.25 = 1/4) and repeating decimals (0.333... = 1/3).
- Irrational Numbers: Cannot be expressed as p/q. Non-terminating, non-repeating decimals. Examples: √2, √3, π, e. Note: √4 = 2 is rational, not irrational.
- Real Numbers: Union of rational and irrational numbers. Every point on the number line is a real number.
- HCF (Highest Common Factor): Largest number that divides two or more numbers exactly. Also called GCD. Used when dividing things into smaller equal groups.
- LCM (Least Common Multiple): Smallest number divisible by two or more numbers. Used for finding common cycles, combining fractions.
Formulas / Key Facts
Relationship between LCM and HCF:
- LCM × HCF = Product of two numbers
- LCM(a,b) × HCF(a,b) = a × b (valid only for two numbers)
Prime Factorization Method:
- HCF = Product of common prime factors with lowest powers
- LCM = Product of all prime factors with highest powers
Divisibility Rules (must memorize):
| Divisor | Rule |
|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) |
| 3 | Sum of digits divisible by 3 |
| 4 | Last two digits divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 7 | Double last digit, subtract from rest; result divisible by 7 |
| 8 | Last three digits divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 10 | Last digit is 0 |
| 11 | Difference of (sum of odd-place digits) and (sum of even-place digits) is 0 or divisible by 11 |
Co-prime Numbers: HCF = 1. Example: 8 and 15 are co-prime.
For three numbers a, b, c:
- LCM × HCF ≠ a × b × c (formula doesn't extend directly)
- Use stepwise method: LCM(a,b,c) = LCM(LCM(a,b), c)
Worked Examples
Example 1: Classification Question: Classify √49, √50, 22/7, and 0.121221222...
Solution:
- √49 = 7 → Rational (perfect square gives integer)
- √50 → Irrational (not a perfect square, non-repeating decimal)
- 22/7 → Rational (expressed as p/q form)
- 0.121221222... → Irrational (non-repeating pattern; each block adds one more 2)
Example 2: Finding HCF and LCM Question: Find HCF and LCM of 36 and 48.
Solution: Step 1: Prime factorization
- 36 = 2² × 3²
- 48 = 2⁴ × 3¹
Step 2: HCF = Common primes with lowest powers
- HCF = 2² × 3¹ = 4 × 3 = 12
Step 3: LCM = All primes with highest powers
- LCM = 2⁴ × 3² = 16 × 9 = 144
Verification: 12 × 144 = 1728 = 36 × 48 ✓
Example 3: Application Problem Question: Three bells ring at intervals of 12, 15, and 20 minutes. If they ring together at 9:00 AM, when will they ring together next?
Solution: Find LCM of 12, 15, and 20.
Prime factorization:
- 12 = 2² × 3
- 15 = 3 × 5
- 20 = 2² × 5
LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60 minutes
Next common ring = 9:00 AM + 60 minutes = 10:00 AM
Example 4: Divisibility Check Question: Is 2574 divisible by 11?
Solution: Digits (left to right): 2, 5, 7, 4
- Odd positions (1st, 3rd): 2 + 7 = 9
- Even positions (2nd, 4th): 5 + 4 = 9
- Difference = 9 - 9 = 0
Since difference is 0, yes, 2574 is divisible by 11.
Common Mistakes
Mistake 1: Thinking √(any number) is irrational. → Fix: Check if it's a perfect square first. √16 = 4 (rational), √17 is irrational.
Mistake 2: Confusing terminating vs. repeating decimals. → Fix: Both are rational. 0.5 (terminating) and 0.333... (repeating) are both p/q. Only non-repeating infinite decimals are irrational.
Mistake 3: Applying LCM × HCF = Product formula to three or more numbers. → Fix: This formula works ONLY for two numbers. For three numbers, calculate LCM and HCF separately using stepwise method.
Mistake 4: Forgetting that 1 is neither prime nor composite. → Fix: Prime numbers start from 2. Also, 2 is the only even prime.
Mistake 5: In divisibility by 4 or 8, testing the entire number instead of last digits. → Fix: For 4, check only last two digits. For 8, check only last three digits.
Quick Reference
- Whole numbers = {0, 1, 2, 3, ...}; Natural numbers start from 1.
- Rational = p/q form (includes terminating and repeating decimals); Irrational = non-repeating, non-terminating.
- HCF × LCM = Product of the two numbers (two numbers only).
- For LCM: take highest powers; For HCF: take lowest powers.
- Divisibility by 6 = check both 2 AND 3.
- Divisibility by 11 = (sum of odd-place digits) − (sum of even-place digits) = 0 or multiple of 11.