UPSC Geo‑Scientist – Quantitative Aptitude
Topic: Number System
Why it matters:
The Number System section tests basic arithmetic, logic and pattern recognition. Mastering it gives you confidence for Arithmetic, Algebra, and Data Interpretation sections in the UPSC exam.
1. Simple Explanation
The Number System deals with the properties of integers, fractions, decimals and irrational numbers. In UPSC, the focus is mainly on:
| Sub‑topic | What you’ll encounter | Typical Question Type |
|---|---|---|
| Integers & Natural Numbers | Whole numbers, negatives, zero | Find the 10th term, etc. |
| Prime & Composite | Divisibility, prime factorisation | Identify primes in a list |
| Factors & Multiples | Common factors, LCM, GCD | Compute LCM of 12 and 18 |
| Perfect Squares & Cubes | Squares & cubes of integers | Identify perfect square |
| Divisibility Rules | Quick checks for 2,3,5,7,9,11 | Determine if 12345 is divisible by 3 |
| Place Value & Decimals | Units, tens, etc. | Find the 3rd digit from right |
| Numeric Patterns & Sequences | Arithmetic, geometric, Fibonacci | Find the next term |
| Modulus & Remainder | Remainders on division | ( 7^4 \mod 5 ) |
| Sum of Digits & Digit Properties | Digit sum, digital root | Sum of digits of 12345 |
2. Key Points (Bullet‑style)
| Area | Important Rules / Formulas |
|---|---|
| Integers | Whole numbers (0,1,2,\dots); natural numbers (1,2,3,\dots) |
| Prime Numbers | Number > 1 with only 1 and itself as divisors. |
| Composite Numbers | Any non‑prime integer > 1. |
| Factors & Multiples | • Factors divide exactly. • Multiples are products of a number and any integer. |
| LCM & GCD | • LCM = smallest common multiple. • GCD = greatest common divisor. |
| Perfect Squares | ( n^2 ) where (n) is an integer. • Last digit patterns: 0,1,4,5,6,9. |
| Perfect Cubes | ( n^3 ). • Last digit patterns: 0,1,8,7,4,5,6,3,2,9. |
| Divisibility Rules | • 2: even last digit. • 3: sum of digits divisible by 3. • 4: last two digits divisible by 4. • 5: last digit 0 or 5. • 6: divisible by 2 and 3. • 7: Rule of 7 (double last digit, subtract from remaining). • 8: last three digits divisible by 8. • 9: sum of digits divisible by 9. • 10: last digit 0. • 11: alternating sum of digits divisible by 11. |
| Place Value | Units (10⁰), Tens (10¹), Hundreds (10²), etc. |
| Decimal & Fraction | Simplify fractions, convert to decimals. |
| Modulus | ( a \mod n = r ) where ( r ) is remainder. |
| Digital Root | Repeated sum of digits until one digit remains. Useful for divisibility by 3 or 9. |
3. Worked Example (All Sub‑topics Combined)
Question
A number (N) satisfies the following conditions:
- (N) is a 4‑digit integer.
- The sum of its digits is 22.
- (N) is divisible by 3 and 11.
- (N) is a perfect square.
Find (N).
Step‑by‑Step Solution
| Step | What to do | Reason |
|---|---|---|
| 1 | Use divisibility by 3: sum of digits = 22 → 22 mod 3 = 1 → not divisible by 3. | Contradiction → no such number. |
| 2 | Check divisibility by 11: Alternating sum of digits must be divisible by 11. | Even if step 1 fails, we can still see if any 4‑digit square satisfies both. |
| 3 | List 4‑digit perfect squares: 1000–9999 → 32²=1024 to 99²=9801. | Generate list. |
| 4 | Filter by digit sum 22: Test each square. | 64²=4096 (sum 19), 65²=4225 (sum 13), 66²=4356 (sum 18), 67²=4489 (sum 25), 68²=4624 (sum 16), 69²=4761 (sum 18), 70²=4900 (sum 13), 71²=5041 (sum 10), 72²=5184 (sum 18), 73²=5329 (sum 19), 74²=5476 (sum 22). |
| 5 | Check 5476: Sum of digits = 5+4+7+6=22 ✅. Divisible by 3? 22 mod 3 = 1 → no. | |
| 6 | Check 5929 (77²): Sum 5+9+2+9=25 → no. | |
| 7 | Check 7396 (86²): Sum 7+3+9+6=25 → no. | |
| 8 | Check 7744 (88²): Sum 7+7+4+4=22 ✅. Divisible by 3? 22 mod 3 = 1 → no. | |
| 9 | Check 8281 (91²): Sum |