Introduction to Number System for RRB Exams
The Number System is the bedrock of Mathematics in almost every competitive exam in India, especially for the Railway Recruitment Board (RRB) exams like NTPC, Group D, and Technician. It is the first chapter in the quantitative aptitude syllabus and serves as the foundation for other topics like HCF-LCM, Simplification, and Algebra. Understanding the nature of numbers, their properties, and the rules governing them is essential for solving complex arithmetic problems quickly and accurately. In this guide, we will break down the Number System into its core components, providing you with the shortcuts and concepts needed to ace the RRB exams.
Topic Weightage and Importance
In RRB NTPC (CBT 1 & 2) and RRB Group D, the Number System holds significant weightage. Based on previous year paper analyses, aspirants can expect 2 to 4 questions directly from this topic. However, its indirect importance is much higher, as concepts like divisibility and unit digits are used to eliminate options in various other chapters. For RRB Technician Grade I and III, the Number System is equally vital as it forms the basis of the 'Number Systems' section in the official syllabus. Mastering this topic ensures a strong start to your preparation and helps in scoring easy marks.
Key Concepts and Formulas
To master the Number System, you must first understand the classification of numbers and certain arithmetic properties.
1. Classification of Numbers
- Natural Numbers (N): Counting numbers starting from 1 (1, 2, 3, ...).
- Whole Numbers (W): Natural numbers including zero (0, 1, 2, ...).
- Integers (Z): All positive and negative whole numbers (..., -2, -1, 0, 1, 2, ...).
- Rational Numbers: Numbers that can be expressed as p/q, where q is not zero.
- Irrational Numbers: Numbers that cannot be expressed as p/q (e.g., √2, π).
- Prime Numbers: Numbers greater than 1 that have only two factors: 1 and itself (2, 3, 5, 7, 11, ...). Note: 2 is the only even prime number.
- Composite Numbers: Numbers that have more than two factors (4, 6, 8, 9, ...).
2. Place Value and Face Value
- Face Value: The value of the digit itself. In 543, the face value of 4 is 4.
- Place Value: The value based on its position (units, tens, hundreds). In 543, the place value of 4 is 40.
3. Divisibility Rules
| Number | Divisibility Rule |
|---|---|
| 2 | Last digit must be even (0, 2, 4, 6, 8). |
| 3 | Sum of digits must be divisible by 3. |
| 4 | Last two digits must be divisible by 4. |
| 5 | Last digit must be 0 or 5. |
| 6 | Must be divisible by both 2 and 3. |
| 8 | Last three digits must be divisible by 8. |
| 9 | Sum of digits must be divisible by 9. |
| 11 | Difference between the sum of digits at odd places and even places must be 0 or a multiple of 11. |
4. Unit Digit Calculation
To find the unit digit of a number raised to a power (a^n), divide 'n' by 4 and use the remainder as the new power. If the remainder is 0, use 4 as the power. Note: Numbers ending in 0, 1, 5, or 6 always have the same unit digit regardless of the power.
Solved Examples (Step-by-Step)
Example 1: Find the unit digit of (234)^102 + (234)^103.
Solution:
1. For (234)^102: The unit digit of the base is 4. Rule for 4: If power is even, unit digit is 6; if odd, it is 4.
2. 102 is even, so unit digit of (234)^102 = 6.
3. 103 is odd, so unit digit of (234)^103 = 4.
4. Sum = 6 + 4 = 10. The unit digit of the sum is 0.
Example 2: What is the smallest number that should be added to 4456 so that the sum is exactly divisible by 6?
Solution:
1. For a number to be divisible by 6, it must be divisible by 2 and 3.
2. 4456 is already even, so it is divisible by 2.
3. Sum of digits of 4456 = 4+4+5+6 = 19.
4. The next multiple of 3 after 19 is 21.
5. Difference = 21 - 19 = 2.
6. Therefore, 2 must be added.
Example 3: If the number 653xy is divisible by 80, then find the value of x+y.
Solution:
1. 80 = 8 x 10. For divisibility by 10, the last digit 'y' must be 0.
2. Now the number is 653x0. For divisibility by 8, the last three digits '3x0' must be divisible by 8.
3. Testing values for x: If x=2, 320/8 = 40 (Divisible). If x=6, 360/8 = 45 (Divisible).
4. Usually, the smallest digit or a specific condition is asked. Let's take x=2. x+y = 2+0 = 2.
Common Mistakes to Avoid
- Confusing Prime and Odd Numbers: Remember that not all odd numbers are prime (e.g., 9, 15) and 2 is a prime number even though it is even.
- Divisibility by 8: Students often check only the last two digits (like for 4) instead of the last three digits.
- Remainder 0 in Unit Digits: When the power is perfectly divisible by 4, many students use 0 as the power. Always use 4 as the power in this case.
- Calculation Errors in 11's Rule: Always double-check the subtraction of 'sum of odd places' and 'sum of even places'.
Practice Questions with Solutions
1. Which of the following is a prime number? (A) 91 (B) 87 (C) 97 (D) 119
2. Find the sum of the first 20 natural numbers.
3. Find the unit digit of 7^95 - 3^58.
4. If 5432*7 is divisible by 9, what is the digit in place of *?
5. Find the difference between the place value and face value of 7 in the number 27650.
Solutions:
1. Answer: (C) 97. 91 = 13x7, 87 = 29x3, 119 = 17x7. 97 has no factors other than 1 and itself.
2. Answer: 210. Formula: [n(n+1)]/2 = [20(21)]/2 = 10 x 21 = 210.
3. Answer: 4. 95/4 leaves remainder 3 (7^3 = 343, unit digit 3). 58/4 leaves remainder 2 (3^2 = 9). 3 - 9 becomes (13 - 9) = 4.
4. Answer: 6. Sum of digits: 5+4+3+2+x+7 = 21 + x. To be divisible by 9, the sum must be 27. x = 27 - 21 = 6.
5. Answer: 6993. Place value = 7000. Face value = 7. Difference = 7000 - 7 = 6993.
Frequently Asked Questions (FAQs)
Q1: Is 1 a prime number?
A1: No, 1 is neither prime nor composite. Prime numbers must have exactly two distinct factors.
Q2: How do I find if a large number is prime?
A2: Take the approximate square root of the number. Check divisibility by all prime numbers less than that square root. If none divide it, the number is prime.
Q3: What is the formula for the sum of squares of first n natural numbers?
A3: The formula is [n(n + 1)(2n + 1)] / 6.
Conclusion and Final Tips
The Number System is the gateway to mastering Mathematics for RRB exams. Focus on learning divisibility rules and the classification of numbers by heart, as they save precious seconds during the exam. Practice finding unit digits and remainders daily to build speed. Remember, consistent practice of basic concepts is what separates a successful candidate from the rest. Keep practicing, and stay confident!
