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

**Number Properties:** • Sum of first n natural numbers = n(n+1)/2 • Sum of first n odd numbers = n² • Sum of first n even numbers = n(n+1) • A perfect square ends only in 0, 1, 4, 5, 6, or 9 (never 2, 3, 7, 8) • The number of zeros at the end of n! = (n/5) + (n/25) + (n/125) + ...

**Co-primes:** Two numbers whose HCF is 1 (e.g., 8 and 15)

**Consecutive Integers:** Product of n consecutive integers is always divisible by n!

Worked Examples

**Example 1: Number Classification** *Question:* Which of the following is a rational number but not an integer? Options: (a) -5 (b) 0 (c) 3/4 (d) √16

*Solution:*

• (a) -5 is an integer, hence also rational
• (b) 0 is an integer, hence also rational
• (c) 3/4 = 0.75, rational but NOT an integer ✓
• (d) √16 = 4, an integer

**Answer: (c) 3/4**

**Example 2: Divisibility Test** *Question:* Is 5,346,732 divisible by 4?

*Solution:* For divisibility by 4, check only the last two digits: 32 32 ÷ 4 = 8 (exactly divisible) Therefore, 5,346,732 is divisible by 4.

**Example 3: Finding Missing Digit** *Question:* The number 6x2 is divisible by 3. Find the smallest value of x.

*Solution:* For divisibility by 3, sum of digits must be divisible by 3. Sum = 6 + x + 2 = 8 + x For smallest x: 8 + x must be divisible by 3 8 + 1 = 9 ✓ (divisible by 3) **Answer: x = 1**

**Example 4: Even-Odd Logic** *Question:* If a and b are odd numbers, what is the nature of (a² + b²)?

*Solution:* Odd² = Odd (e.g., 3² = 9, 5² = 25) So a² = odd, b² = odd Odd + Odd = Even **Answer: (a² + b²) is always even**

Common Mistakes

**Mistake 1:** Confusing "natural numbers start from 0" → **Correct:** Natural numbers start from 1; whole numbers start from 0.

**Mistake 2:** Treating 1 as prime → **Correct:** 1 is neither prime nor composite by definition. The smallest prime is 2.

**Mistake 3:** Testing divisibility by 6 using only the last digit → **Correct:** A number divisible by 6 must satisfy both divisibility by 2 AND by 3 simultaneously.

**Mistake 4:** Assuming all square roots are irrational → **Correct:** √4 = 2, √9 = 3, √16 = 4 are rational. Only non-perfect square roots are irrational (√2, √3, √5...).

**Mistake 5:** Forgetting negative integers when counting integers → **Correct:** Integers include {..., -2, -1, 0, 1, 2, ...}, not just positive whole numbers.

Quick Reference

✓ **Every natural number is a whole number; every whole number is an integer; every integer is a rational number.**

✓ **Divisibility by 3 or 9: sum the digits. For 11: alternate sum difference.**

✓ **2 is the only even prime; all other primes are odd.**

✓ **Rational numbers include all integers, fractions, terminating and repeating decimals.**

✓ **Even ± Even = Even; Odd ± Odd = Even; Even × Anything = Even.**

✓ **Perfect squares end in 0, 1, 4, 5, 6, or 9 only.**