Numbers

Overview

Numbers form the foundation of mathematics and are a core component of the AP TET Mathematics section. This topic tests your understanding of whole numbers, integers, place value systems, divisibility rules, and methods for finding HCF and LCM—concepts that primary school teachers must explain clearly to young learners.

For AP TET Paper I (Classes 1-5) and Paper II (Classes 6-8), expect 3-5 questions directly from this topic. Questions typically test computational skills, conceptual understanding of number properties, and the ability to apply divisibility rules quickly. Mastery here also supports related topics like fractions, decimals, and arithmetic operations.

Students must be comfortable with both procedural calculations (finding HCF/LCM) and conceptual questions (why a number is divisible by certain factors). Speed and accuracy in mental math will be crucial.

Key Concepts

• **Natural numbers** are counting numbers starting from 1 (1, 2, 3, ...). **Whole numbers** include zero along with natural numbers (0, 1, 2, 3, ...).

• **Integers** extend whole numbers to include negative numbers (..., -3, -2, -1, 0, 1, 2, 3, ...). Every whole number is an integer, but not every integer is a whole number.

• **Place value** assigns a value to a digit based on its position. In 5,847, the digit 5 has place value 5,000 (thousands place), while face value remains 5.

• **Divisibility** means one number divides another exactly without remainder. If a divides b exactly, we write a | b.

• **Factors** are numbers that divide a given number exactly. **Multiples** are products of a number with natural numbers.

• **HCF (Highest Common Factor)** is the largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).

• **LCM (Least Common Multiple)** is the smallest number that is divisible by two or more given numbers.

• **Relationship between HCF and LCM**: For any two numbers a and b, HCF(a, b) × LCM(a, b) = a × b.

Formulas / Key Facts

**Place Value System (Indian):** | Place | Value | |-------|-------| | Units | 1 | | Tens | 10 | | Hundreds | 100 | | Thousands | 1,000 | | Ten Thousands | 10,000 | | Lakhs | 1,00,000 | | Ten Lakhs | 10,00,000 | | Crores | 1,00,00,000 |

**Divisibility Rules:**

• **By 2**: Last digit is even (0, 2, 4, 6, 8)
• **By 3**: Sum of digits is divisible by 3
• **By 4**: Last two digits form a number divisible by 4