NUMBER SYSTEM
CSAT
Number System is one of the most fundamental and high-utility topics of CSAT (General Studies Paper II). Questions from this area are generally simple in appearance but test conceptual clarity, logical thinking, and calculation efficiency. A strong command over Number System helps not only in direct questions but also in topics like Percentage, Ratio, Time & Work, Data Interpretation, and Logical Reasoning.
In OPSC Prelims, number system questions are usually:
• Concept-based rather than formula-heavy
• Designed to trap careless calculation
• Easy to moderate in difficulty
Hence, mastering basics ensures safe scoring with minimum time investment.
CLASSIFICATION OF NUMBERS
2.1 Natural Numbers
• Numbers used for counting objects
• Start from 1 and go up to infinity
• Do not include zero or negative numbers
Example: 1, 2, 3, 4, …
2.2 Whole Numbers
• Natural numbers along with zero
• Represent complete quantities
Example: 0, 1, 2, 3, 4, …
2.3 Integers
• Include all positive numbers, negative numbers, and zero
• Represent gains, losses, temperature changes, etc.
Example: … -3, -2, -1, 0, 1, 2, 3 …
2.4 Rational Numbers
• Numbers that can be expressed in the form p/q
• p and q are integers and q ≠ 0
• Includes:
– Terminating decimals (0.25, 0.5)
– Recurring decimals (0.333…, 0.666…)
– Fractions (3/4, -7/5)
2.5 Irrational Numbers
• Cannot be written in p/q form
• Decimal expansion is non-terminating and non-repeating
• Example:
– √2
– √3
– π
2.6 Real Numbers
• Combination of rational and irrational numbers
• All numbers used in everyday calculations
EVEN AND ODD NUMBERS
Even Numbers
• Divisible by 2
• Last digit is 0, 2, 4, 6, or 8
Odd Numbers
• Not divisible by 2
• Last digit is 1, 3, 5, 7, or 9
Important Properties
• Even + Even = Even
• Odd + Odd = Even
• Even + Odd = Odd
• Even × Any Number = Even
• Odd × Odd = Odd
PRIME AND COMPOSITE NUMBERS
Prime Numbers
• Numbers greater than 1
• Have exactly two distinct factors: 1 and itself
• Example: 2, 3, 5, 7, 11, 13
Important Facts
• 2 is the only even prime number
• Prime numbers are building blocks of all composite numbers
Composite Numbers
• Numbers having more than two factors
• Example: 4, 6, 8, 9, 10, 12
Special Case
• 1 is neither prime nor composite
FACTORS AND MULTIPLES
Factor
• A number that divides another number exactly
• Example: Factors of 12 → 1, 2, 3, 4, 6, 12
Multiple
• A number obtained by multiplying a given number by integers
• Example: Multiples of 5 → 5, 10, 15, 20 …
Key Difference
• Factors are finite
• Multiples are infinite
PRIME FACTORISATION
• Any composite number can be expressed as a product of prime numbers
• Example:
36 = 2 × 2 × 3 × 3
Uses
• Finding HCF and LCM
• Simplifying fractions
• Solving divisibility problems
HIGHEST COMMON FACTOR (HCF)
Meaning
• Greatest number that divides two or more numbers exactly
Methods
1. Prime factorisation method
2. Division method
Key Properties
• HCF of two numbers always divides their difference
• HCF is always less than or equal to the smallest number
LEAST COMMON MULTIPLE (LCM)
Meaning
• Smallest number exactly divisible by two or more numbers
Key Formula
• For two numbers:
LCM × HCF = Product of the two numbers
DIVISIBILITY RULES
Divisibility by 2
• Last digit is even
Divisibility by 3
• Sum of digits divisible by 3
Divisibility by 4
• Last two digits divisible by 4
Divisibility by 5
• Last digit 0 or 5
Divisibility by 6
• Divisible by both 2 and 3
Divisibility by 8
• Last three digits divisible by 8
Divisibility by 9
• Sum of digits divisible by 9
Divisibility by 10
• Last digit 0
Divisibility by 11
• Difference between sum of digits in odd and even places is divisible by 11 or zero
REMAINDER CONCEPT
• Remainder is always less than divisor
• If a number is completely divisible, remainder is zero
Important Concepts
• (a + b) mod n = [(a mod n) + (b mod n)] mod n
• Used in large number and power-based questions
LAST DIGIT OF A NUMBER
• Only unit digit affects the last digit of result
• Cyclic patterns exist in powers
Example
• Powers of 2 → 2, 4, 8, 6 (cycle of 4)
• Powers of 3 → 3, 9, 7, 1
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Subject: CSAT
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