Number System – Complete Short Notes for Competitive Exams

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1. Basics of Numbers

• Natural Numbers (N): (1, 2, 3, 4, …)

• Whole Numbers (W): (0, 1, 2, 3, …)

• Integers (Z): (..., -3, -2, -1, 0, 1, 2, 3, …)

• Rational Numbers (Q): Numbers that can be expressed as (p/q), where (p,q \in Z), (q \neq 0)

• Irrational Numbers: Cannot be expressed as a fraction, e.g., (\sqrt{2}, \pi)

• Real Numbers (R): All rational + irrational numbers

• Prime Numbers: Greater than 1, divisible only by 1 and itself

• Composite Numbers: More than 1, divisible by numbers other than 1 and itself

• Co-prime Numbers: Numbers whose HCF = 1

• Even & Odd Numbers: Even divisible by 2, odd not divisible by 2

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2. Properties of Numbers

1. Closure:

   • Addition, multiplication of integers → integer

   • Not closed under subtraction & division always

2. Commutativity:

   • (a+b = b+a), (a \times b = b \times a)

3. Associativity:

   • ((a+b)+c = a+(b+c)), ((a \times b) \times c = a \times (b \times c))

4. Distributivity:

   • (a \times (b+c) = a \times b + a \times c)

5. Identity:

   • Additive identity: 0

   • Multiplicative identity: 1

6. Inverse:

   • Additive inverse of (a) is (-a)

   • Multiplicative inverse of (a) is (1/a), (a \neq 0)

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3. Divisibility Rules

• 2: Last digit even

• 3: Sum of digits divisible by 3

• 4: Last two digits divisible by 4

• 5: Last digit 0 or 5

• 6: Divisible by 2 & 3

• 7: Double last digit, subtract from remaining → divisible by 7

• 8: Last three digits divisible by 8

• 9: Sum of digits divisible by 9

• 10: Last digit 0

• 11: Alternating sum of digits divisible by 11

• 12: Divisible by 3 & 4

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4. HCF & LCM

• HCF (Highest Common Factor): Largest number dividing all numbers

• LCM (Least Common Multiple): Smallest number divisible by all numbers

• Formulas:

  • (HCF \times LCM = a \times b) (for 2 numbers)

• Methods:

  • Prime Factorization

  • Division Method

Shortcut:

• HCF of consecutive numbers = 1

• LCM of co-prime numbers = product of numbers

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5. Conversion Between Bases

• Binary (2), Octal (8), Decimal (10), Hexadecimal (16)

• Decimal → Any Base: Divide repeatedly by base, read remainders bottom to top

• Any Base → Decimal: Multiply digits by powers of base

• Binary → Octal/Hexadecimal: Group digits in 3 (Octal) or 4 (Hex)

Example:

• (101101_2 = 45_{10})

• (101101_2 = 55_8)

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6. Fractions & Decimals

• Fractions: (p/q), q ≠ 0

• Simplest Form: p and q co-prime

• Recurring Decimal → Fraction:

  • Example: (0.\overline{3} = 1/3)

• Terminating Decimal → Fraction: Denominator of form (2^m5^n)

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7. Square & Cube Properties

• Perfect Squares: 1,4,9,16,…

• Perfect Cubes: 1,8,27,64,…

• Shortcut: Last digit pattern for squares & cubes

  • Squares end in 0,1,4,5,6,9

  • Cubes end in 0-9: 0→0,1→1,2→8,3→7,4→4,5→5,6→6,7→3,8→2,9→9

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8. Special Numbers

• Armstrong Number: Sum of cubes of digits = number

• Palindrome Number: Same forward & backward

• Automorphic Number: Square ends with the number itself

• Perfect Number: Sum of factors excluding number = number

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9. Modular Arithmetic / Remainders

• Basic Formula:

  • ((a+b) \mod n = [(a \mod n) + (b \mod n)] \mod n)

  • ((a \times b) \mod n = [(a \mod n) \times (b \mod n)] \mod n)

• Divisibility Shortcut: For large numbers, sum of digits, alternate digits, etc.

Example:

• (123456 \mod 9 = 1+2+3+4+5+6 = 21 → 2+1 = 3)

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10. Tricks & Tips

1. Check divisibility before large calculation

2. Use factorization for LCM & HCF quickly

3. Remainders pattern repeats (modular arithmetic)

4. Memorize squares up to 30 & cubes up to 20

5. Use last digits to solve power problems

   • (7^{2025}), last digit → pattern of 7: 7,9,3,1 repeat every 4

6. Co-prime numbers → fast LCM

7. Sum of n natural numbers: (S = n(n+1)/2)

8. Sum of squares: (S = n(n+1)(2n+1)/6)

9. Sum of cubes: (S = [n(n+1)/2]^2)

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11. Quick Examples

1. Find HCF & LCM of 24 & 36:

   • Prime factors: (24=2^33), (36=2^23^2)

   • HCF = (2^23=12), LCM = (2^33^2=72)

2. Find remainder: (2^{2025} \mod 5)

   • Pattern of 2^n mod 5 → 2,4,3,1 repeat every 4

   • 2025 mod 4 = 1 → remainder = 2

3. Convert decimal 156 → binary:

   • 156 ÷2=78 r0, 78 ÷2=39 r0, 39 ÷2=19 r1, 19 ÷2=9 r1, 9 ÷2=4 r1, 4 ÷2=2 r0, 2 ÷2=1 r0, 1 ÷2=0 r1

   • Binary = 10011100

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12. One-Liner Formulas for Quick Revision

• Sum of n natural numbers = (n(n+1)/2)

• Sum of squares = (n(n+1)(2n+1)/6)

• Sum of cubes = ([n(n+1)/2]^2)

• HCF × LCM = Product of two numbers

• Last digit patterns:

  • Square: 0→0,1→1,2→4,3→9,4→6,5→5,6→6,7→9,8→4,9→1

  • Cube: 0→0,1→1,2→8,3→7,4→4,5→5,6→6,7→3,8→2,9→9

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✅ Exam Day Strategy:

• Always check divisibility first

• Use last digit / remainder tricks for powers

• Memorize squares, cubes, prime numbers

• For HCF & LCM, factorization beats division method for speed