Number System – Quantitative Aptitude
For Nagaland TET / IIT‑JEE style competitive exams
Objective – Master the fundamentals of the number system, derive key properties and formulas, recognize common pitfalls, and solve tough problems.
1. The Building Blocks
| Category | Definition | Example | Key Property |
|---|---|---|---|
| Integers | Positive, negative, or zero | … –3, –2, –1, 0, 1, 2, 3 … | Closed under +, – and × |
| Rational Numbers | Ratio of two integers, denominator ≠ 0 | ½, –4/7, 3 | Closed under +, – and × |
| Irrational Numbers | Cannot be expressed as a ratio of integers | √2, π | Closed under +, – and × (except division by zero) |
| Prime Numbers | Positive integers > 1 with exactly two distinct divisors: 1 and itself | 2, 3, 5, 7, 11 | Every integer > 1 is either prime or a product of primes |
| Composite Numbers | Positive integers > 1 that are not prime | 4, 6, 8, 9 | Have at least one divisor other than 1 and itself |
| Perfect Square / Cube | Integer that is a square / cube of another integer | 9 = 3², 27 = 3³ | Useful for simplifying radicals |
| Perfect Number | Sum of its proper divisors equals the number | 6, 28 | Not needed for most TET problems but good to know |
| Palindromic Number | Reads the same forward and backward | 121, 1331 | Useful in digit‑wise problems |
2. Fundamental Properties & Theorems
| Property | Statement | Quick Proof / Insight |
|---|---|---|
| Associativity | (a + b) + c = a + (b + c); (a × b) × c = a × (b × c) | Follows from definition of addition/multiplication on integers |
| Commutativity | a + b = b + a; a × b = b × a | Axiom of the integers |
| Distributivity | a × (b + c) = a × b + a × c | Axiom of the integers |
| Transitivity of Divisibility | If a | b and b |
| Euclidean Algorithm | For any a, b ∈ ℤ⁺, there exist q, r with a = bq + r, 0 ≤ r < b | Gives gcd(a, b) = gcd(b, r) |
| Fundamental Theorem of Arithmetic | Every integer > 1 has a unique prime factorisation (up to order) | Key to GCD/LCM formulas |
| LCM × GCD Property | lcm(a, b) × gcd(a, b) = a × b | Derived from prime factorizations |
| Modular Arithmetic | a ≡ b (mod n) ⇔ n | (a – b) |
| Chinese Remainder Theorem (brief) | For pairwise coprime moduli nᵢ, a unique solution modulo N = n₁n₂… exists | Useful for contest problems |
3. Common Operations & Tricks
| Operation | Shortcut / Trick | When to Use |
|---|---|---|
| Divisibility Rules | 2 – even; 3 – digit sum divisible by 3; 4 – last two digits divisible by 4; 5 – ends in 0/5; 6 – divisible by 2 & 3; 7 – double last digit, subtract from remaining part… | Quick checks before long division |
| Powers & Roots | aⁿ × aᵐ = aⁿ⁺ᵐ ; (aⁿ)ᵐ = aⁿᵐ ; a⁰ = 1 ; a^(–n) = 1/aⁿ | Simplify expressions |
| Modular Exponentiation | aⁿ mod m can be found by repeated squaring | Avoids huge numbers |
| Sum of G.P. | Sₙ = a (rⁿ – 1)/(r – 1) if r ≠ 1 | Useful for “sum of powers” problems |
| Sum of Arithmetic Series | Sₙ = n/2 (first + last) | Quick for palindromic digit sums |
| Chinese Remainder | Combine congruences stepwise | Solve simultaneous remainders |
| Prime Factorisation of Large Numbers | Use trial division up to √N or use known factorisation patterns | For GCD/LCM of huge numbers |
4. Common Pitfalls & Tricky Points
| Pitfall | Why It Occurs | How to Avoid |
|---|---|---|
| Negative Exponents | Misinterpreting a^(–n) as –aⁿ | Remember a^(–n) = 1/aⁿ |
| Fractional Exponents | Confusing √(a²) with a vs. | √(a²) = |
| Modular Arithmetic with Negative Numbers | Writing –3 mod 5 as 2 without justification | Use 0 ≤ remainder < modulus: –3 mod 5 = 2 |
| Divisibility by 9 vs 3 | Mistaking digit sum rule for 9 | For 9, sum digits; if divisible by 9 → number divisible by 9 |
| Large Powers | Writing 2¹⁰⁰ incorrectly | Use binary exponentiation or logarithms |
| **Misapplying LCM × GCD |