SSC (CGL/CHSL) – Quantitative Aptitude
Topic: Number System
Why this topic matters?
The SSC exams test your conceptual clarity and speed on basic properties of numbers—integers, primes, divisibility, modular arithmetic, powers & roots, and decimal expansions. A solid grasp of these fundamentals saves time and avoids pitfalls.
1. Quick Reference Table
| Concept | Symbol | Key Property/Rule | Example |
|---|---|---|---|
| Natural Numbers | ℕ | 1,2,3,… | 5 ∈ ℕ |
| Whole Numbers | ℤ⁺ ∪ {0} | 0,1,2,… | 0 ∈ ℤ⁺ |
| Integers | ℤ | All positive, negative & zero | –3 ∈ ℤ |
| Rational Numbers | ℚ | Fraction of integers | ⅓ ∈ ℚ |
| Irrational Numbers | ℝ \ ℚ | No finite fraction representation | √2 ∈ ℝ |
| Prime | P | Only divisible by 1 and itself | 13 ∈ P |
| Composite | C | >1 with at least one proper divisor | 12 ∈ C |
| Power | aⁿ | a multiplied by itself n times | 2⁵ = 32 |
| Modulus | a mod b | Remainder when a divided by b | 17 mod 5 = 2 |
| Floor | ⌊x⌋ | Greatest integer ≤ x | ⌊2.7⌋ = 2 |
| Ceiling | ⌈x⌉ | Smallest integer ≥ x | ⌈2.3⌉ = 3 |
2. Types of Numbers & Their Basic Properties
| Class | Definition | Key Properties |
|---|---|---|
| Integers | ℤ = { …, −3, −2, −1, 0, 1, 2, 3, … } | Closure under +, –, ×. Associative, Commutative, Distributive. |
| Rational | ℚ = { a/b | a,b ∈ ℤ, b ≠ 0 } |
| Irrational | ℝ \ ℚ | Not expressible as a ratio of integers. |
| Reals | ℝ = ℚ ∪ (ℝ\ℚ) | Ordered field. |
| Complex | ℂ = ℝ + iℝ | i² = –1. |
Tricky Point – Zero is an integer but not a prime or composite.
3. Divisibility Rules (Quick‑look Cheat Sheet)
| Divisible By | Rule | Example |
|---|---|---|
| 2 | Even last digit | 128 is divisible by 2 |
| 3 | Sum of digits divisible by 3 | 123 → 1+2+3=6 → 6 mod 3 = 0 |
| 4 | Last two digits divisible by 4 | 316 → 16 is divisible by 4 |
| 5 | Last digit 0 or 5 | 305 |
| 6 | Divisible by 2 & 3 | 42 |
| 7 | Trick: Double last digit, subtract from remaining prefix, repeat until small | 203 → 20 – 2×3 = 14 → 1–4×? (common in exam) |
| 8 | Last three digits divisible by 8 | 1040 → 040 → 40 |
| 9 | Sum of digits divisible by 9 | 729 → 7+2+9=18 → 1+8=9 |
| 10 | Last digit 0 | 110 |
Note: For 7, 11, 13, 17, 19, there are recursive tricks; master a few to save time.
4. Prime Numbers & Factorization
4.1 Prime Factorization (PF)
Every integer ( n > 1 ) can be written uniquely as
[ n = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k} ]
where ( p_i ) are distinct primes and ( a_i ) are positive integers.
- Utility: Compute GCD, LCM, divisors, sum of divisors.
4.2 Sum of Divisors Function (\sigma(n))
If ( n = \prod p_i^{a_i} ),
[ \sigma(n) = \prod_{i=1}^{k} \frac{p_i^{a_i+1}-1}{p_i-1}