Nagaland TET – Quantitative Aptitude
Topic: Number System
The “Number System” section tests a candidate’s ability to understand and manipulate different kinds of numbers (integers, fractions, decimals, primes, etc.), recognise patterns, and perform basic arithmetic operations. Below are clear, concise notes, key points, a worked example, and a quick revision summary to help you master this topic.
1. Simple Explanation
| Concept | What it Means | Quick Example |
|---|---|---|
| Integers | Whole numbers including positives, negatives, and zero. | -3, 0, 7 |
| Natural Numbers | Positive integers starting from 1. | 1, 2, 3, … |
| Whole Numbers | Natural numbers + 0. | 0, 1, 2, … |
| Rational Numbers | Numbers that can be expressed as a fraction p/q (p, q integers, q ≠ 0). | ½, 4/3, –5 |
| Irrational Numbers | Numbers that cannot be expressed as a simple fraction (non‑terminating, non‑repeating decimals). | √2, π |
| Prime Number | A positive integer greater than 1 that has no positive divisors other than 1 and itself. | 2, 3, 5, 7, 11 |
| Composite Number | A positive integer that has divisors other than 1 and itself. | 4, 6, 8 |
| Perfect Square / Cube | Numbers that are squares or cubes of integers. | 16 (=4²), 27 (=3³) |
| Odd / Even | Numbers not divisible by 2 are odd; divisible by 2 are even. | 5 – odd, 8 – even |
| Divisibility Rules | Quick tests to decide if one number divides another without long division. | 9 → sum of digits divisible by 9 |
| LCM / GCD | Least Common Multiple and Greatest Common Divisor – essential for simplifying fractions and solving equations. | LCM(4,6)=12, GCD(18,24)=6 |
| Number Bases | Representation of numbers in bases other than 10 (binary, octal, hexadecimal). | binary 1010₂ = decimal 10 |
| Conversion | Changing a number from one base to another. | 1100₂ → 12₁₀ |
2. Key Points (Bullet‑style)
| Category | Key Points |
|---|---|
| Fundamental Types | • Integers, rationals, irrationals. • Natural, whole, positive, negative. |
| Prime & Composite | • Prime test: check divisibility up to √n. • Any integer >1 is either prime or composite. |
| Divisibility Rules | • 2: last digit even. • 3 & 9: sum of digits divisible by 3/9. • 4: last two digits divisible by 4. • 5: last digit 0 or 5. • 6: divisible by 2 & 3. • 7: (double last digit subtracted from rest) divisible by 7. • 8: last three digits divisible by 8. • 10: last digit 0. |
| LCM & GCD | • GCD via Euclid’s algorithm. • LCM = (a × b)/GCD(a,b). • For more numbers: pairwise GCDs or prime factorisation. |
| Factorisation | • Write n = p₁^a₁ × p₂^a₂ × … • Count of factors = (a₁+1)(a₂+1)… |
| Perfect Powers | • Square: n = k². • Cube: n = k³. • Check using integer roots. |
| Number Bases | • Binary (base‑2), Octal (base‑8), Hex (base‑16). • Conversion to/from decimal: multiply digits by powers of base. |
| Decimals & Fractions | • Terminating decimal ⇔ denominator’s prime factors are only 2 and 5. • Repeating decimal ⇔ other primes in denominator. |
| Patterns & Sequences | • Arithmetic progression (AP): a, a+d, a+2d… • Geometric progression (GP): a, ar, ar²… • Recognise common difference/ratio quickly. |
| Special Numbers | • Fibonacci numbers, triangular numbers, etc., may appear in puzzles. |
3. Worked Example
Problem:
Find the least common multiple (LCM) and greatest common divisor (GCD) of the numbers 48, 180, 210.
Then simplify the fraction 48/180 using the GCD.
Step 1: Prime Factorisation
| Number | Factorisation |
|---|---|
| 48 | 2⁴ × 3 |
| 180 | 2² × 3² × 5 |
| 210 | 2 × 3 × 5 × 7 |
Step 2: GCD (Greatest Common Divisor)
Take the minimum exponent for each common prime:
- Prime 2: min(4, 2, 1) = 1 → 2¹
- Prime 3: min(1, 2, 1) = 1 → 3¹