Number System Formulas – 60-Second Revision
Number System – One-Pager
Key Points (One-Liners)
- Natural numbers: 1, 2, 3 … ∞ → N
- Whole numbers: 0 included → W = N ∪ {0}
- Integers: … –2, –1, 0, 1, 2 … → Z
- Rational: p/q (q ≠ 0, p,q ∈ Z); decimal either terminating or repeating.
- Irrational: non-terminating & non-repeating decimals → π, √2, e.
- Real numbers = Rational ∪ Irrational → R
- Prime: exactly two factors; 2 is the only even prime.
- Co-prime: HCF = 1 (e.g. 8 & 15).
- LCM × HCF = product of two numbers.
- For fractions a/b & c/d: (a/b) ÷ (c/d) = (a×d)/(b×c).
- Successive division: if N divided by a, b, c leaves remainders r₁, r₂, r₃, then N = k·LCM(a,b,c) + final remainder.
- Divisibility by 11: (sum of digits at odd places) – (sum at even places) = 0 or multiple of 11.
- To find unit digit of aⁿ, cycle of last digit repeats every 4 (for bases 2–9).
- Binary → decimal: Σ bit×2ⁿ; decimal → binary: divide-by-2 collect remainders reverse.
Important Formulas / Rules
| Formula / Rule | When to Use |
|---|---|
| 1. HCF(a,b) × LCM(a,b) = a×b | Two-number problems |
| 2. LCM of fractions = LCM(numerators) / HCF(denominators) | Fraction LCM |
| 3. HCF of fractions = HCF(numerators) / LCM(denominators) | Fraction HCF |
| 4. Remainder when (a×b×c) ÷ m = [(a mod m)(b mod m)(c mod m)] mod m | Product remainder |
| 5. Euler: a^φ(m) ≡ 1 (mod m) if gcd(a,m)=1 | Large power remainder |
| 6. (a + b)² = a² + 2ab + b² | Algebra & quick squares |
| 7. a³ + b³ = (a + b)(a² – ab + b²) | Sum of cubes |
| 8. a³ – b³ = (a – b)(a² + ab + b²) | Difference of cubes |
| 9. Sum of first n odd numbers = n² | Pattern recognition |
| 10. 1 + 2 + … + n = n(n+1)/2 | Arithmetic series |
Memory Tricks
- “L-H Product”: LCM & HCF always multiply to give the product of the two numbers.
- “Even-Prime Lonely 2”: only even prime → 2 (visualise 2 standing alone).
- “11 Jump-Frog”: for 11 divisibility, let digits jump over neighbour and subtract.
- “4-Cycle Last Digit”: unit digits repeat every 4 powers → divide exponent by 4, use remainder.
- “LCM tops, HCF bottoms”: when dealing with fractions, LCM takes numerators’ LCM over denominators’ HCF (and vice-versa for HCF).
Common Mistakes
| Mistake | Correct Approach |
|---|---|
| Treating 1 as prime | 1 has only one factor; primes need exactly two. |
| Ignoring 0 in whole vs natural | 0 is whole but NOT natural. |
| Cancelling digits in repeating decimals | 0.333… ≠ 0.33; use fraction 1/3. |
| Forgetting to reverse remainders in binary conversion | Collect remainders bottom-up. |
| Applying LCM formula directly to 3+ numbers without checking pairwise HCF | Use prime factor method or successive LCM. |
Last-Minute Tips
- Rule-of-9 check: for quick calc verification, cast out 9s.
- Unit digit first: answer options differ in last digit → find unit digit only, save time.
- Cancel before multiply: reduces fraction work & avoids big numbers.
- Prime-table till 50: recall 2-3-5-7-11-13-17-19-23-29-31-37-41-43-47.
- 60-second scan: glance for keywords “remainder”, “divisibility”, “LCM”, “HCF” → pick the matching formula straight away.
Quick Practice (5 MCQs)
1. What is the least 3-digit number exactly divisible by 3, 5 and 7?
LCM(3,5,7)=105. Ans: 105
2. If 3⁷⁵ is divided by 10, the remainder is?
Unit digit of 3 cycles 3-9-7-1. 75 mod 4 = 3 → 7. Remainder = 7
3. HCF of 2/3, 4/9 and 5/6 is?
HCF(2,4,5)=1; LCM(3,9,6)=18 → HCF = 1/18
4. Sum of first 20 odd numbers equals?
n² = 20² = 400
5. How many prime numbers between 50 and 60?
53, 59 → 2
BETA
