Number System
⚡ Unlock time-saving calculation tricks and mental math techniques. Solve complex problems in seconds with proven shortcut methods used by top performers.
Shortcut Techniques
Study MaterialShortcut Techniques – Number System
Shortcut techniques in Number System help candidates solve aptitude questions quickly and accurately in competitive exams like SSC, Banking, Railway, CAT, CDS, UPSC, and campus placements.
Instead of lengthy calculations, these techniques focus on:
- Fast divisibility checks
- Mental calculations
- Pattern recognition
- Approximation methods
- Remainder shortcuts
- Unit digit tricks
- Algebraic simplifications
Why Learn Shortcut Techniques?
- Reduces calculation time significantly.
- Improves speed in online exams.
- Helps solve difficult questions mentally.
- Useful for elimination techniques in MCQs.
- Improves overall quantitative aptitude performance.
Shortcut #1: Divisibility Rules
Divisibility rules help determine whether a number is divisible by another number without performing actual division.
| Divisor | Shortcut Rule |
|---|---|
| 2 | Last digit should be 0, 2, 4, 6, or 8 |
| 3 | Sum of digits should be divisible by 3 |
| 4 | Last two digits should be divisible by 4 |
| 5 | Last digit should be 0 or 5 |
| 6 | Number should be divisible by both 2 and 3 |
| 8 | Last three digits should be divisible by 8 |
| 9 | Sum of digits should be divisible by 9 |
| 10 | Last digit should be 0 |
| 11 | Difference of alternate digit sums should be divisible by 11 |
✔ Divisibility rules are among the most frequently used shortcuts in aptitude exams.
Shortcut #2: Unit Digit Tricks
In many aptitude questions, only the unit digit is required. Instead of calculating the entire value, focus only on the last digits.
Important Patterns
| Number | Unit Digit Pattern |
|---|---|
| 2n | 2, 4, 8, 6 (repeats) |
| 3n | 3, 9, 7, 1 (repeats) |
| 4n | 4, 6 (repeats) |
| 5n | Always 5 |
| 6n | Always 6 |
| 7n | 7, 9, 3, 1 (repeats) |
| 8n | 8, 4, 2, 6 (repeats) |
| 9n | 9, 1 (repeats) |
Example: Find the unit digit of 723
Pattern of 7 = 7, 9, 3, 1
23 ÷ 4 leaves remainder 3
Third number in the pattern = 3
Shortcut #3: Fast Square Calculation
Use algebraic identities to calculate squares quickly.
Formula
(a − b)² = a² + b² − 2ab
Example: 297²
= (300 − 3)²
= 300² + 3² − 2 × 300 × 3
= 90000 + 9 − 1800
= 88209
Shortcut #4: Cube Identities
Cube identities are useful for simplification and algebra-based aptitude problems.
- a³ + b³ = (a + b)(a² − ab + b²)
- a³ − b³ = (a − b)(a² + ab + b²)
✔ Frequently asked in simplification and algebraic aptitude questions.
Shortcut #5: Recurring Decimal to Fraction Conversion
Recurring decimals can quickly be converted into fractions using direct formulas.
Basic Rule
For:
0.ababab...
Fraction = Repeating Digits / Corresponding 9s
| Recurring Decimal | Fraction |
|---|---|
| 0.333... | 1/3 |
| 0.232323... | 23/99 |
| 0.125125... | 125/999 |
Shortcut #6: Remainder Tricks
Remainder questions are very common in competitive exams.
Important Property
Dividend = (Divisor × Quotient) + Remainder
Example:
Find the remainder when 6767 + 67 is divided by 68.
67 ≡ -1 (mod 68)
(-1)67 + 67
= -1 + 67
= 66
Shortcut #7: Digital Sum Technique
Digital sum helps in checking divisibility and simplifying calculations.
Method
Add digits repeatedly until a single digit is obtained.
| Number | Digital Sum |
|---|---|
| 5678 | 5 + 6 + 7 + 8 = 26 → 2 + 6 = 8 |
| 9999 | 9 + 9 + 9 + 9 = 36 → 3 + 6 = 9 |
✔ Useful in divisibility by 3 and 9.
Shortcut #8: Multiplication Near Base Values
When numbers are close to powers of 10, calculations become faster using approximation techniques.
Example: 98 × 97
= (100 − 2)(100 − 3)
= 10000 − 300 − 200 + 6
= 9506
Shortcut #9: Even-Odd Property Tricks
- Even + Even = Even
- Odd + Odd = Even
- Even + Odd = Odd
- Even × Any Number = Even
- Odd × Odd = Odd
✔ Very useful in option elimination.
Shortcut #10: Prime Number Techniques
Prime numbers greater than 3 are generally in the form:
6n ± 1
Examples:
5 = 6(1) − 1
7 = 6(1) + 1
11 = 6(2) − 1
Important Exam Tips
- Memorize divisibility rules properly.
- Learn squares up to 50 and cubes up to 20.
- Practice unit digit patterns daily.
- Use approximation wherever possible.
- Avoid lengthy calculations in MCQs.
- Focus on remainder and simplification tricks.
- Use algebraic identities to reduce solving time.
Shortcut techniques in Number System help candidates improve calculation speed, reduce mistakes, and solve aptitude questions efficiently in competitive examinations.