Number System — Study Notes for UP Police Constable

Overview

The Number System forms the mathematical foundation for almost every numerical question you'll encounter in the UP Police Constable exam. This topic typically accounts for 3–5 direct questions and underlies problems in simplification, HCF-LCM, divisibility tests, and remainder problems. Mastery here means you can quickly identify number types, apply divisibility rules, and manipulate prime factorization—all essential for speed and accuracy under exam pressure.

Understanding number classifications (natural, whole, integer, rational) allows you to eliminate wrong answer choices instantly. Divisibility rules help you skip lengthy division in multiple-choice questions. Number properties like even-odd arithmetic, prime-composite distinction, and perfect square/cube recognition are repeatedly tested in series, analogy, and calculation problems.

The key to success is pattern recognition and mental math shortcuts. Rather than memorizing hundreds of facts, focus on the core properties and practice applying them to mixed problems.

Key Concepts

• **Natural Numbers (N)**: Counting numbers starting from 1: {1, 2, 3, 4, ...}. Used for counting discrete objects; no zero, no negatives, no fractions.

• **Whole Numbers (W)**: Natural numbers plus zero: {0, 1, 2, 3, ...}. The smallest whole number is 0; there is no largest whole number.

• **Integers (Z)**: All whole numbers and their negative counterparts: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Includes positive integers, negative integers, and zero.

• **Rational Numbers (Q)**: Any number expressible as p/q where p and q are integers and q ≠ 0. Includes terminating decimals (0.5 = 1/2) and repeating decimals (0.333... = 1/3).

• **Prime vs Composite**: A prime number has exactly two distinct factors: 1 and itself (2, 3, 5, 7, 11...). Composite numbers have more than two factors (4, 6, 8, 9...). Note: 1 is neither prime nor composite; 2 is the only even prime.

• **Even and Odd Properties**: Even ± Even = Even; Odd ± Odd = Even; Even ± Odd = Odd. For multiplication: Even × Anything = Even; Odd × Odd = Odd.

• **Divisibility**: A number 'a' is divisible by 'b' if a ÷ b leaves zero remainder. Divisibility rules provide shortcuts to test this without actual division.

• **Place Value System**: In a number like 5,342, the digit 5 represents 5,000 (place value), while its face value is simply 5. Understanding this helps in digit-based problems and number formation questions.

Formulas / Key Facts

**Divisibility Rules:** • **By 2**: Last digit is 0, 2, 4, 6, or 8 • **By 3**: Sum of all digits is divisible by 3 • **By 4**: Last two digits form a number divisible by 4 • **By 5**: Last digit is 0 or 5 • **By 6**: Divisible by both 2 and 3 • **By 8**: Last three digits form a number divisible by 8 • **By 9**: Sum of all digits is divisible by 9 • **By 10**: Last digit is 0 • **By 11**: Difference between sum of digits at odd places and sum at even places is 0 or divisible by 11