Number System
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Quantitative Aptitude - Number System
Master the Fundamentals. Score the Easy Marks. Win the Section.
1. What Is the Number System?
The Number System is the foundation of Quantitative Aptitude. It includes:
- Classification of numbers (Natural, Integers, etc.)
- Divisibility & remainders
- Factors, multiples
- LCM, HCF
- Powers, unit digits
- Base conversions
- Trailing zeros in factorials
2. Types of Numbers
| Type | Examples | Notes |
|---|---|---|
| Natural | 1, 2, 3, ... | Counting numbers |
| Whole | 0, 1, 2, 3, ... | Natural + 0 |
| Integers | ..., -3, -2, -1, 0, 1, 2, ... | No decimals |
| Rational | 2/3, 0.75 | Can be written as \( \frac{p}{q} \) |
| Irrational | \( \pi \), \( \sqrt{2} \) | Non-repeating, non-terminating |
| Prime | 2, 3, 5, 7, 11... | Only divisible by 1 and itself |
| Composite | 4, 6, 8... | More than 2 factors |
Tip: Memorize prime numbers up to 50. 1 is neither prime nor composite.
3. Divisibility Rules
- By 2: Last digit even
- By 3: Sum of digits divisible by 3
- By 4: Last two digits divisible by 4
- By 5: Ends in 0 or 5
- By 6: Divisible by 2 and 3
- By 9: Sum of digits divisible by 9
- By 11: Difference of alternating digit sums divisible by 11
Exam Tip: Eliminate options quickly using these rules.
4. HCF and LCM
Use the identity:
\( \text{HCF} \times \text{LCM} = \text{Product of Numbers} \)
Apply via prime factorization or ladder method.
5. Factors and Multiples
- Total number of factors of \( N = p^a \cdot q^b \cdot r^c \) is \( (a+1)(b+1)(c+1) \)
- Use Venn diagrams or prime factor trees for clarity.
6. Remainders and Modular Arithmetic
Apply the identity:
\( a = nq + r \Rightarrow a \mod n = r \)
Use power cycles and Euler’s/Fermat’s theorems for large powers.
Example: \( 7^{103} \mod 5 \): Cycle is 7, 9, 3, 1 → 103 mod 4 = 3 → Answer = 3
7. Unit Digits of Powers
Find repeating cycle of last digits:
- \( 2^n \): 2, 4, 8, 6 (repeat every 4)
- \( 3^n \): 3, 9, 7, 1
- \( 7^n \): 7, 9, 3, 1
Use \( n \mod \text{cycle length} \) to find position.
8. Trailing Zeros in Factorials
Formula:
\( \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \cdots \)
Example: 100! → 100/5 = 20, 100/25 = 4 → Total = 24 zeros
9. Base Conversions
- Decimal to Binary: Divide by 2 repeatedly, reverse the remainders.
- Binary to Decimal: Multiply digits by \( 2^n \) and sum.
10. Exam Strategy and Speed Tips
- Eliminate options using divisibility & patterns
- Memorize power cycles of 2-9
- Use modular arithmetic to reduce large power problems
- For factorial zeros, only count powers of 5
- Practice 20-25 questions per sub-topic before exam
11. Final Revision Checklist
- Divisibility rules (2-11)
- Prime numbers till 50
- Power cycles (2 to 9)
- Factor count formulas
- Remainder tricks and mod rules
Summary: Understand the logic, learn the shortcuts, and practice with time limits. That’s the formula to crack Number System questions in competitive exams.
Topics
- Number System
- H.C.F. & L.C.M.
- Decimal Fractions
- Simplification
- Square Roots & Cube Roots
- Average
- Problems on Numbers
- Problems on Ages
- Surds & Indices
- Percentage
- Operations on Numbers
- Profit & Loss
- Ratio & Proportion
- Partnership
- Chain Rule
- Time & Work
- Pipes & Cistern
- Time & Distance
- Problems on Trains
- Boats & Streams
- Alligation or Mixture
- Simple Interest
- Compound Interest
- Logarithms
- Area
- Volume & Surface Areas
- Mensuration
- Races & Games of Skill
- Calendar
- Clocks
- Stocks & Shares
- Permutations & Combinations
- Probability
- True Discount
- Banker’s Discount
- Heights & Distances
- Odd Man Out & Series
- Algebra
- Divisibility Rules & Remainder Theorem
- Factorials & Power Cycles
- Progressions (A.P., G.P., H.P.)
- Work & Wages
- Set Theory / Venn Diagrams
- Geometry (Lines, Angles, Triangles, Polygons)
- Coordinate Geometry
- Data Interpretation
- Trigonometry
- Statistics (Mean, Median, Mode)
- Arithmetic Operations
- Quadratic Equations