🚀 Number System Quick Revision – RRB
🔑 Key Concepts
- Natural (N): 1, 2, 3 …
- Whole (W): 0, 1, 2 …
- Integers (Z): … –2, –1, 0, 1 …
- Rational: p/q (q ≠ 0) → decimal may terminate/repeat
- Irrational: decimal never repeats/terminates (√2, π)
- Real = Rational + Irrational
- Even: 2k; Odd: 2k±1
- Prime: >1, divisible only by 1 & itself; Co-prime: HCF = 1
- Twin primes: (3,5), (5,7)…; Composite: >1 & not prime
| Rule | Formula / Trick |
|---|
| Sum of 1st n natural numbers | n(n+1)/2 |
| Sum of 1st n odd numbers | n² |
| Sum of 1st n even numbers | n(n+1) |
| Divisibility by 3 | Digit-sum divisible by 3 |
| Divisibility by 9 | Digit-sum divisible by 9 |
| Divisibility by 11 | (sum odd-place digits – sum even-place digits) = 0 or multiple of 11 |
| Last digit of powers (cyclicity) | 2: 2,4,8,6; 3: 3,9,7,1; 4: 4,6; 7: 7,9,3,1; 8: 8,4,2,6; 9: 9,1 |
| HCF × LCM | = Product of two numbers |
| Remainder of (a×b)÷m | ≡ (remainder a ÷m × remainder b ÷m) mod m |
⚡ Shortcuts & Tricks
- Unit digit in 3⁴⁷ → 47 mod 4 = 3 → 3³ = 7
- Difference of squares: a²–b² = (a–b)(a+b) → quick mental calc.
- Square near 50: (50±x)² = 2500 ±100x + x²
- Square near 100: (100±x)² = 10000 ±200x + x²
- Digital root (9-rule): for quick divisibility & cross-checking sums
- Count of zeros in n!: ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ …
- Any power of 5 ends with 5; even power of 6 ends with 6
🧠 Mnemonics
- “1 2 3” → All primes >3 are of form 6k ±1
- “PPP” → Prime Pair Points: (3,5), (5,7), (11,13)… twin primes
- “0-2-4-6-8” → Even digits; rest odd
- “9-Sum-Dance” → For 3 & 9, dance with digit-sum
📝 Common Exam Q & A
Q1: Find unit digit of 4¹⁵⁷ × 7²⁵.
4 cycle: 4,6… 157 odd → 4; 7 cycle: 7,9,3,1… 25 mod4=1 → 7. Unit digit = 4×7 = 8.Q2: Is 57123 divisible by 11?
(3+1+5)–(2+7)=9–9=0 → Yes.Q3: LCM of 24 & 36?
24=2³·3, 36=2²·3² → LCM=2³·3²=72.Q4: Remainder when 2⁴⁰ divided by 5?
Cycle of 2 mod5: 2,4,3,1. 40 mod4=0 → 1.
⚡ Quick Facts Table
| Type | First Member | Example | Test Trick |
|---|
| Prime | 2 | 97 | Not divisible by any prime ≤ √n |
| Co-prime pair | (4,15) | GCD=1 | Check HCF=1 |
| Perfect square | 1 | 144 | Unit digit 0,1,4,5,6,9 & digital root 1,4,7,9 |
| Perfect cube | 1 | 125 | Digital root 1,8,9,0 |
| Terminating decimal | 5/8=0.625 | Denominator only 2 &/or 5 factors | |
| Repeating decimal | 1/3=0.3̅ | Denominator has other prime factor | |
✅ Last-Day Checklist
All the best—may your calc be fast & options faster!
BETA
