Number System
Overview
Number System forms the backbone of Quantitative Aptitude in IBPS Clerk Prelims. While direct questions (1–2 per exam) may seem few, the concepts—HCF, LCM, divisibility, factors, and surds—underpin simplification, data interpretation, and word problems throughout the paper. Mastering this topic gives you speed advantages across multiple question types.
For Clerk-level exams, expect straightforward applications: finding HCF/LCM of 2–3 numbers, divisibility-based shortcuts, factor counting, and basic surd simplification. The difficulty stays moderate, but mistakes here cascade into wrong answers elsewhere. Focus on building quick mental-math reflexes rather than memorizing obscure rules.
Key Concepts
- HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Used when you need to "reduce" or find the greatest common measure (e.g., cutting ropes into equal pieces).
- LCM (Least Common Multiple): The smallest number divisible by all given numbers. Used when events repeat or align (e.g., bells ringing together, circular track meetings).
- HCF × LCM = Product of two numbers: This identity holds only for exactly two numbers—a frequent exam trap.
- Prime Factorization: Express a number as a product of primes (e.g., 60 = 2² × 3 × 5). This unlocks HCF, LCM, and factor-count problems instantly.
- Divisibility Rules: Shortcuts to check if a number divides another without actual division. Critical for simplification speed.
- Co-prime Numbers: Two numbers with HCF = 1. If HCF(a, b) = 1, then LCM(a, b) = a × b.
- Surds: Irrational roots that cannot be simplified to a rational number (e.g., √2, ∛5). Rationalization removes surds from denominators.
- Number of Factors Formula: If N = p^a × q^b × r^c, then total factors = (a+1)(b+1)(c+1).
Formulas / Key Facts
| Concept | Formula / Rule |
|---|---|
| HCF × LCM | HCF(a,b) × LCM(a,b) = a × b (for two numbers only) |
| LCM of fractions | LCM of numerators ÷ HCF of denominators |
| HCF of fractions | HCF of numerators ÷ LCM of denominators |
| Number of factors | N = p^a × q^b → Factors = (a+1)(b+1) |
| Sum of factors | (p^0 + p^1 + ... + p^a)(q^0 + ... + q^b)... |
| Divisibility by 2 | Last digit is even (0, 2, 4, 6, 8) |
| Divisibility by 3 | Sum of digits divisible by 3 |
| Divisibility by 4 | Last two digits divisible by 4 |
| Divisibility by 5 | Last digit is 0 or 5 |
| Divisibility by 6 | Divisible by both 2 and 3 |
| Divisibility by 8 | Last three digits divisible by 8 |
| Divisibility by 9 | Sum of digits divisible by 9 |
| Divisibility by 11 | Difference of sum of alternate digits divisible by 11 |
| Rationalizing surds | Multiply by conjugate: a + √b → a − √b |
| √a × √b | = √(ab) |
| √a ÷ √b | = √(a/b) |
Worked Examples
Example 1: HCF and LCM Application
Find the HCF and LCM of 36 and 48.
Step 1: Prime factorize both numbers.
- 36 = 2² × 3²
- 48 = 2⁴ × 3¹
Step 2: HCF = Product of lowest powers of common primes.
- HCF = 2² × 3¹ = 4 × 3 = 12
Step 3: LCM = Product of highest powers of all primes.
- LCM = 2⁴ × 3² = 16 × 9 = 144
Verification: HCF × LCM = 12 × 144 = 1728 = 36 × 48 ✓
Example 2: Number of Factors
How many factors does 180 have?
Step 1: Prime factorize 180.
- 180 = 2² × 3² × 5¹
Step 2: Apply factor formula.
- Total factors = (2+1)(2+1)(1+1) = 3 × 3 × 2 = 18
Answer: 180 has 18 factors.
Example 3: Surd Rationalization
Simplify: 6 ÷ (√3 + √2)
Step 1: Multiply numerator and denominator by conjugate (√3 − √2).
- = 6(√3 − √2) ÷ [(√3 + √2)(√3 − √2)]
Step 2: Denominator becomes difference of squares.
- = 6(√3 − √2) ÷ (3 − 2)
- = 6(√3 − √2) ÷ 1
Answer: 6√3 − 6√2
Example 4: Divisibility Check
Is 7524 divisible by 12?
For divisibility by 12, check both 3 and 4.
Divisibility by 3: Sum of digits = 7 + 5 + 2 + 4 = 18. Since 18 ÷ 3 = 6, divisible by 3 ✓
Divisibility by 4: Last two digits = 24. Since 24 ÷ 4 = 6, divisible by 4 ✓
Answer: Yes, 7524 is divisible by 12.
Common Mistakes
- Using HCF × LCM = Product for three or more numbers → This formula works only for exactly two numbers. For three numbers, use prime factorization separately for HCF and LCM.
- Confusing when to use HCF vs LCM in word problems → "Greatest/largest that divides" signals HCF. "Smallest that is divisible by" or "events occurring together" signals LCM.
- Forgetting +1 in factor formula → If 60 = 2² × 3 × 5, factors = (2+1)(1+1)(1+1) = 12, not 2 × 1 × 1 = 2. Each exponent needs +1.
- Incomplete rationalization → After multiplying by conjugate, students forget to simplify the denominator. Always compute (a+b)(a−b) = a² − b² completely.
- Applying wrong divisibility rule → Divisibility by 4 checks last TWO digits, by 8 checks last THREE digits. Students often check just the last digit for both.
Quick Reference
- HCF: Take lowest powers; LCM: Take highest powers of all primes
- HCF × LCM = Product (only for 2 numbers)
- Factors of N = p^a × q^b → (a+1)(b+1)
- Divisibility by 11: (Sum of odd-position digits) − (Sum of even-position digits) divisible by 11
- Rationalize: Multiply top and bottom by conjugate surd
- Co-primes: HCF = 1, so LCM = direct product