Number System

Overview

The Number System forms the foundation of all arithmetic and quantitative reasoning in TS TET Paper I and II. This topic tests your understanding of how numbers are classified, represented, and manipulated—skills essential for teaching primary mathematics effectively.

Expect 3–5 direct questions on whole numbers, integers, place value, divisibility rules, and HCF/LCM calculations. Beyond direct questions, a strong grasp of number properties helps you solve problems across fractions, algebra, and word problems faster. Mastery here means understanding not just procedures but *why* they work—crucial for explaining concepts to young learners.

Focus areas: quick application of divisibility tests, efficient HCF/LCM computation, and confident handling of negative integers.

Key Concepts

• **Natural Numbers (N)**: Counting numbers starting from 1. N = {1, 2, 3, 4, ...}. Zero is *not* included.

• **Whole Numbers (W)**: Natural numbers plus zero. W = {0, 1, 2, 3, ...}. Every natural number is a whole number.

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

• **Place Value vs Face Value**: In 4,725—the face value of 7 is always 7; its place value is 700 (7 × 100). Place value depends on position; face value does not.

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

• **Factors and Multiples**: Factors divide a number exactly; multiples are products of a number with natural numbers. Example: Factors of 12 are {1, 2, 3, 4, 6, 12}; multiples of 12 are {12, 24, 36, ...}.

• **HCF (Highest Common Factor)**: The largest number that divides two or more numbers exactly. Also called GCD.

• **LCM (Least Common Multiple)**: The smallest number that is a multiple of two or more numbers.

Formulas / Key Facts

| Concept | Rule / Formula | |---------|----------------| | Divisibility by 2 | Last digit is 0, 2, 4, 6, or 8 | | Divisibility by 3 | Sum of digits is divisible by 3 | | Divisibility by 4 | Last two digits form a number divisible by 4 | | Divisibility by 5 | Last digit is 0 or 5 | | Divisibility by 6 | Divisible by both 2 and 3 | | Divisibility by 8 | Last three digits form a number divisible by 8 | | Divisibility by 9 | Sum of digits is divisible by 9 | | Divisibility by 11 | Difference between sum of odd-place digits and sum of even-place digits is 0 or divisible by 11 | | HCF × LCM | HCF(a, b) × LCM(a, b) = a × b (valid for two numbers) | | Number of factors | If n = p^a × q^b × r^c, then total factors = (a+1)(b+1)(c+1) |