Karnataka Police SI (Quantitative Aptitude) – Number System
Quick Study Guide – 5 minutes of revision
1️⃣ Simple Explanation
The number system is a way of writing numbers using a set of symbols (digits) and a positional base (radix).
- Decimal (Base‑10) – the one we use every day (0–9).
- Other common systems:
- Binary (Base‑2) – (0, 1)
- Octal (Base‑8) – (0–7)
- Hexadecimal (Base‑16) – (0–9, A–F)
- In positional notation, the k‑th digit from the right represents the number of base^k units.
Why it matters:
- Many SI problems involve conversions, divisibility rules, or pattern detection that rely on understanding these systems.
2️⃣ Key Points
| Topic | What to Know | Tips |
|---|---|---|
| Place Value | digit × base^position |
Write a number in expanded form to check calculations. |
| Conversion (Dec ↔ Bin/Oct/Hex) | Repeated division for Dec→Base, repeated multiplication for Base→Dec | Remember the remainders (for Dec→Base) or the digits (for Base→Dec). |
| Divisibility Rules | 2,3,4,5,6,8,9,10,11,12,13,15,16,20,25,50,100 | Use the sum of digits, last digits, or patterns. |
| Factors & Multiples | LCM, GCD, prime factorization | Useful for “smallest number divisible by given set”. |
| Modular Arithmetic | a ≡ b (mod m) |
Helps in problems like “last digit of 3^100”. |
| Common Mistakes | Forgetting base in conversion, mis‑reading digits (e.g., A=10), wrong remainder order |
Double‑check by converting back to decimal. |
3️⃣ Conversion Techniques
3.1 Decimal → Base‑b (b = 2, 8, 16, …)
- Divide the decimal number by b.
- Record the remainder (digit in the target base).
- Replace the number by the quotient and repeat.
- Stop when quotient = 0.
- Read the digits from last remainder to first.
Example: Convert 156 (decimal) to binary.
| Step | Quotient | Remainder |
|---|---|---|
| 1 | 156 ÷ 2 = 78 | 0 |
| 2 | 78 ÷ 2 = 39 | 0 |
| 3 | 39 ÷ 2 = 19 | 1 |
| 4 | 19 ÷ 2 = 9 | 1 |
| 5 | 9 ÷ 2 = 4 | 1 |
| 6 | 4 ÷ 2 = 2 | 0 |
| 7 | 2 ÷ 2 = 1 | 0 |
| 8 | 1 ÷ 2 = 0 | 1 |
Binary: 10011100₂
3.2 Base‑b → Decimal
- Write the number in expanded form:
( \sum \text{digit}_k \times b^k ). - Add the terms.
Example: 1011₁₀ (binary) → Decimal.
(1×2^3 + 0×2^2 + 1×2^1 + 1×2^0 = 8 + 0 + 2 + 1 = 11_{10}).
4️⃣ Operations in Different Bases
| Operation | Binary | Octal | Hexadecimal |
|---|---|---|---|
| Addition | Add digits, carry 1 (when sum ≥ 2) | Carry 1 (when sum ≥ 8) | Carry 1 (when sum ≥ 16) |
| Subtraction | Borrow 1 (add 2 to the lower digit) | Borrow 1 (add 8) | Borrow 1 (add 16) |
| Multiplication | Shift left for powers of 2 | Shift left for powers of 8 | Shift left for powers of 16 |
| Division | Shift right / long division | Same | Same |
Tip: For mental math, use binary addition as a quick check for even/odd properties: a number is odd iff its least‑significant bit (LSB) is 1.
5️⃣ Common Problem Types
- Smallest number divisible by a set of integers
→ Compute LCM via prime factorization. - Divisibility checks
→ Apply rules; e.g., a number is divisible by 9 if sum of digits is 9, 18, etc. - Converting between bases
→ Use repeated division/multiplication. - Finding the last digit of a large power
→ Use cyclicity of residues: e.g., powers of 3 mod 10 cycle every 4. - Digit sum patterns
→ Useful for “digital root” problems.
6️⃣ Worked Example (Full Solution)
Question
Find the smallest positive integer that is divisible by 4, 6, 8, and 9.