Number System Guide & Practice

1. Fundamentals & Definitions

• Natural Numbers: Counting numbers starting from 1. (e.g., 1, 2, 3, ...)
• Whole Numbers: Natural numbers including zero. (e.g., 0, 1, 2, 3, ...)
• Integers: All whole numbers and their negative counterparts. (e.g., ..., -2, -1, 0, 1, 2, ...)
• Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero.
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., √2, π).
• Real Numbers: All rational and irrational numbers.
• Even Numbers: Integers divisible by 2.
• Odd Numbers: Integers not divisible by 2.
• Prime Numbers: Numbers greater than 1 having exactly two factors: 1 and the number itself. (e.g., 2, 3, 5, 7, 11, ...). Note: 2 is the only even prime number.
• Composite Numbers: Numbers greater than 1 that have more than two factors. (e.g., 4, 6, 8, 9, 10, ...)
• Co-prime Numbers: Two numbers whose Highest Common Factor (HCF) is 1. (e.g., 15 and 28).
• Face Value: The actual value of the digit itself. (In 5832, the face value of 8 is 8).
• Place Value: The value of a digit based on its position in the number. (In 5832, the place value of 8 is 800).
• Factor: A number that divides another number exactly.
• Multiple: The product of a number with an integer.

2. Core Concepts & Formulas

Divisibility Rules

HCF and LCM

• HCF (Highest Common Factor): The greatest number that divides two or more numbers exactly.
• LCM (Least Common Multiple): The smallest number that is divisible by two or more numbers.
• Fundamental Relation: For any two numbers, Product of the numbers = HCF × LCM.
• Co-primes: Product of co-prime numbers = Their LCM.

Factors (Factorization)

• If a number N can be expressed as a product of its prime factors as N = a^p × b^q × c^r, where a, b, c are prime numbers, then:
  • Total number of factors = (p+1)(q+1)(r+1)

Remainders and Cyclicity

• Remainder: The amount left over after a division. A negative remainder can be made positive by adding the divisor.
• Cyclicity of Unit Digits: The unit digits of powers of a number repeat in a cycle.
  • Cycle of 2: 2, 4, 8, 6 (length 4)
  • Cycle of 3: 3, 9, 7, 1 (length 4)
  • Cycle of 4: 4, 6 (length 2)
  • Cycle of 7: 7, 9, 3, 1 (length 4)
  • Cycle of 8: 8, 4, 2, 6 (length 4)
• To find the unit digit of x^y, find the remainder of y when divided by the cycle length of x.

Fractions and Decimals

• Terminating Decimal: A fraction results in a terminating decimal if its denominator, after simplification, has only prime factors of 2 and/or 5.
• Recurring Decimal: A decimal in which one or more digits repeat indefinitely.

Key Formulas