Math Mastery for Competitive Exams
Welcome, aspirants! Speed and accuracy are the twin pillars of cracking any competitive examination (SSC, Bank, Railways, etc.). This guide breaks down two of the most powerful calculation tools: Digital Sum and Unit Digits.
Topic 1: Complete Concept of Digital Sum
The Digital Sum (DS) is the sum of all the digits of a number until it is reduced to a single digit. In mathematics, the Digital Sum of a number is exactly equal to the remainder left when that number is divided by 9.
The Golden Rule: "Casting Out Nines"
Since the Digital Sum is based on the divisibility rule of 9, you can completely ignore the digit 9 or any digits that add up to 9 while calculating. This makes calculations lightning fast.
Example: Find the DS of 72945
- Standard Method: 7 + 2 + 9 + 4 + 5 = 27 → 2 + 7 = 9
- Ninja Method: Ignore (7+2), ignore 9, ignore (4+5). Remaining is 0. (Note: In DS, 0 and 9 are treated as the same thing). So, DS = 9.
Uses in Calculations (Option Elimination)
The Digital Sum of the LHS (Left Hand Side) of an equation will ALWAYS equal the Digital Sum of the RHS. We use this to verify options without doing the full calculation.
1. Addition Verification
Question: 4352 + 1243 + 2315 = ?
Options: (a) 7910 (b) 7920 (c) 7810 (d) 7900
Solution: Calculate the DS of the question.
- DS of 4352: 4+5=9 ignore, 3+2=5. DS is 5.
- DS of 1243: 1+2+4+3=10 → 1+0=1. DS is 1.
- DS of 2315: 2+3+1+5=11 → 1+1=2. DS is 2.
Total DS: 5 + 1 + 2 = 8. Now find the option with a DS of 8.
Option (a) 7+9+1+0 → Ignore 9, 7+1=8. Answer is (a).
2. Multiplication Check
Question: 85 × 132 = ?
Options: (a) 11210 (b) 11220 (c) 11240
Solution: Check DS for both numbers.
- DS of 85: 8+5=13 → 4.
- DS of 132: 1+3+2 = 6.
Total DS: 4 × 6 = 24 → 2+4 = 6. Now check options.
Option (b) 11220: 1+1+2+2+0 = 6. Answer is (b).
3. The Trick with Subtraction
Sometimes, when subtracting, the DS becomes negative. Rule: Simply add 9 to any negative DS to make it positive.
Example: DS is -4. The real DS is -4 + 9 = 5.
4. How to Handle Division
Division is tricky because the denominator must be converted to a DS of 1. You do this by multiplying both the numerator and denominator by a specific number.
- If denominator DS is 2, multiply by 5 (2×5=10 → DS 1)
- If denominator DS is 5, multiply by 2
- If denominator DS is 4, multiply by 7 (4×7=28 → DS 1)
- If denominator DS is 7, multiply by 4
- If denominator DS is 8, multiply by 8 (8×8=64 → DS 1)
Exceptions & Limitations of Digital Sum
Never use Digital Sum under the following conditions:
- When calculating approximate values (it only works for exact calculations).
- When the denominator's Digital Sum is 3, 6, or 9. (You must first simplify the fraction to remove the factor of 3).
- When multiple options have the exact same Digital Sum. (In this case, use Unit Digits!).
Topic 2: Complete Concept of Unit Digits
The Unit Digit is the rightmost digit of a number (the digit in the "ones" place). In competitive exams, you are often asked to find the unit digit of massive expressions like (247)153. This is where the concept of Cyclicity comes in.
What is Cyclicity?
The unit digits of the powers of a number repeat in a predictable pattern. This repeating pattern is called the cyclicity of that number.
Example: 21=2, 22=4, 23=8, 24=16 (unit digit 6), 25=32 (unit digit 2 again). The cycle is 4 long.
The Three Golden Rules of Cyclicity
Rule 1: Cyclicity of 1 (Numbers 0, 1, 5, 6)
These numbers are stubborn. No matter what power you raise them to, their unit digit never changes.
- (...0)n = 0
- (...1)n = 1
- (...5)n = 5
- (...6)n = 6
Example: Find the unit digit of (156)348. Because it ends in 6, the unit digit is simply 6.
Rule 2: Cyclicity of 2 (Numbers 4 and 9)
These numbers alternate between two unit digits based on whether the power is odd or even.
| Number ending in | If Power is Odd | If Power is Even |
|---|---|---|
| 4 | 4 (e.g., 41=4) | 6 (e.g., 42=16) |
| 9 | 9 (e.g., 91=9) | 1 (e.g., 92=81) |
Example: Find the unit digit of (289)145. The number ends in 9. The power 145 is odd. Therefore, the unit digit is 9.
Rule 3: Cyclicity of 4 (Numbers 2, 3, 7, 8)
These numbers cycle every 4th power. The method is simple:
- Divide the given power by 4 to find the remainder (R).
- If R = 1, use power 1. If R = 2, use power 2. If R = 3, use power 3.
- If the power is perfectly divisible by 4 (R = 0), use the 4th power.
| Number | R = 1 | R = 2 | R = 3 | R = 0 (Power 4) |
|---|---|---|---|---|
| 2 | 2 | 4 | 8 | 6 |
| 3 | 3 | 9 | 7 | 1 |
| 7 | 7 | 9 | 3 | 1 |
| 8 | 8 | 4 | 2 | 6 |
Question: Find the unit digit of (137)245.
Step 1: The base ends in 7.
Step 2: Divide the power 245 by 4. (Trick: just divide the last two digits, 45, by 4). 45 ÷ 4 gives a remainder of 1.
Step 3: The unit digit will be 71 = 7.
Using Unit Digits in Equations
To find the unit digit of complex equations like (12)34 × (43)56 + (5)12, simply find the unit digit of each term separately and perform the operation just on those single digits!
