Number System

This guide breaks down the essential points from your Number System PDF, making it easy to understand for competitive exams. It includes definitions, formulas, worked examples, cheatsheets, visuals, and practice questions with solutions.

1. What is a Number System?

A number system is a method of representing numbers using a set of symbols and rules. It’s used for counting, measuring, and labeling. The most common systems are:

• Decimal (Base 10): Digits 0–9 (used in daily life and mathematics)
• Binary (Base 2): Digits 0, 1 (used in computers)
• Octal (Base 8): Digits 0–7
• Hexadecimal (Base 16): Digits 0–9, A–F

Visual: Number System Bases

2. Types of Numbers

Natural Numbers (N): 1, 2, 3, 4, … Whole Numbers (W): 0, 1, 2, 3, … Integers (I): …, -3, -2, -1, 0, 1, 2, 3, … Even Numbers: 2, 4, 6, … Odd Numbers: 1, 3, 5, … Prime Numbers: 2, 3, 5, 7, … Composite Numbers: 4, 6, 8, 9, … Co-primes: Two numbers with HCF 1 (e.g., 7 and 9) Rational Numbers: Numbers expressible as p/q, q ≠ 0 Irrational Numbers: Numbers not expressible as p/q (e.g., √2, π) Real Numbers: All rational and irrational numbers Imaginary Numbers: Numbers involving √(-1), denoted as i (e.g., 2i, 7i)¹².

3. Face Value and Place Value

• Face Value: The digit itself (e.g., face value of 7 in 38786 is 7)
• Place Value: The value of the digit based on its position (e.g., place value of 7 in 38786 is 700)¹.

4. Fractions and Decimals

• Proper Fraction: Numerator < Denominator (e.g., 2/3)
• Improper Fraction: Numerator > Denominator (e.g., 5/3)
• Mixed Fraction: Combination of whole and fraction (e.g., 1 2/3)
• Decimal: Fraction with denominator as a power of 10 (e.g., 1.5, 2.75)
• Recurring Decimal: Digits repeat after the decimal (e.g., 0.333…)¹.

5. Key Formulas Cheat Sheet

Number Series Sums

Algebraic Identities

• $ (a+b)^2 = a^2 + 2ab + b^2 $
• $ (a-b)^2 = a^2 - 2ab + b^2 $
• $ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $
• $ (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 $
• $ (a+b)(a-b) = a^2 - b^2 $³².

6. Divisibility Rules (with Examples)

7. Special Number Types

• Perfect Number: Sum of factors (excluding itself) equals the number (e.g., 6, 28)
• Twin Primes: Pair of primes differing by 2 (e.g., 3 & 5)
• Co-primes: No common factor except 1 (e.g., 6 & 35)¹.

8. Factorials and Counting Zeros

• Factorial: $ n! = n \times (n-1) \times … \times 1 $
• Number of zeros in n!: Count number of 5s in prime factors
  • Formula: $ Number of zeros = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + … $
• Example: Number of zeros in 100! is 24 (from 20 + 4)¹.

9. Unit Digit Tricks

• The unit digit of powers cycles (e.g., unit digit of 2ⁿ repeats every 4: 2, 4, 8, 6)
• For products, multiply only unit digits
• Example: Unit digit of 13²⁴ × 68⁵⁷ + 24¹³ × 57⁶⁸ + 1234 is 6¹.

10. Remainder Theorems

• Dividend = Divisor × Quotient + Remainder
• For negative remainders: Negative Remainder = Remainder – Divisor
• For powers: If a number leaves r as remainder when divided by n, its square leaves r² mod n as remainder¹.

11. Practice Questions with Solutions

Q1. If 49¹⁵ – 1 is exactly divisible by: Solution: 49¹⁵ – 1 = (7²)¹⁵ – 1 = 7³⁰ – 1. For even powers, aⁿ – bⁿ is divisible by a – b. Answer: 6

Q2. Find the remainder when 29⁴⁷ + 17⁴⁷ is divided by 46. Solution: For odd n, aⁿ + bⁿ is divisible by a + b. 29 + 17 = 46. Answer: 0

Q3. Find the unit digit of 13²⁴ × 68⁵⁷ + 24¹³ × 57⁶⁸ + 1234. Solution: Unit digit is 6.

Q4. Find the unit digit of 278⁹²³⁵! + 222⁹²³⁵! + 666⁹²³⁵! Solution: Unit digit is 8.

Q5. Find the number of zeros at the end of the product 25! × 32! × 45!. Solution: 6 + 7 + 10 = 23 zeros.

Q6. Find the number of zeros at the end of 41×42×43…..100 Solution: 24 (zeros in 100!) – 9 (zeros in 40!) = 15

Q7. If 5724A is divisible by 11 then find the value of A. Solution: (5+2+A)-(7+4)=A-6=0 ⇒ A=6

Q8. When a natural number N is divided by 3, remainder 1; N+1 divided by 5, remainder 0. Find N. Solution: N=64

Q9. A number when divided by 136 leaves 46 as remainder. If divided by 34, remainder? Solution: 46 divided by 34 leaves 12.

Q10. Two numbers when divided by a divisor give remainders 473 and 298; their sum divided by same divisor gives remainder 236. Find divisor. Solution: 473+298-236=535

12. Visuals and Graphics

Number Line (Integers and Real Numbers):

<--- -3 -- -2 -- -1 -- 0 -- 1 -- 2 -- 3 --->

Venn Diagram: Types of Numbers

[All Numbers]
 ├─ [Real Numbers]
 │   ├─ [Rational Numbers]
 │   │   ├─ [Integers]
 │   │   │   ├─ [Whole Numbers]
 │   │   │   │   ├─ [Natural Numbers]
 │   │   │   └─ [Negative Integers]
 │   │   └─ [Fractions]
 │   └─ [Irrational Numbers]
 └─ [Imaginary Numbers]

Place Value Table (Indian System):

13. Tips and Tricks

• For divisibility, memorize the rules table above.
• For zeros in factorials, count the 5s in the factorization.
• For unit digits, find the power cycle.
• For remainders, use the remainder theorem and modular arithmetic.

14. Summary Cheatsheet

15. Practice More

• Use the PDF’s practice questions above for revision.
• Draw number lines and place value tables for visualization.
• Make your own cheatsheet for formulas and rules.

By mastering these concepts, formulas, and tricks, you’ll be well-prepared for any number system question in competitive exams!