Number System

Whole Numbers, Integers, Place Value, Factors and Multiples

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Overview

The Number System forms the absolute foundation of primary mathematics and is heavily tested in Bihar TET Paper I. This topic checks whether you understand how numbers are structured, how they relate to each other, and whether you can apply basic operations correctly. Expect 3–5 direct questions from this area.

For the exam, you must be comfortable with the hierarchy of number types (natural → whole → integers), place value up to crores, and the relationship between factors, multiples, LCM and HCF. Many questions combine these concepts—for instance, asking you to find common multiples of two numbers or identify place value in a large numeral. A clear mental model of how numbers are classified and manipulated will help you solve problems quickly and avoid silly errors.

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Key Concepts

• **Natural Numbers (N)**: Counting numbers starting from 1. Set = {1, 2, 3, 4, ...}. Zero is NOT a natural number.

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

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

• **Place Value vs Face Value**: Face value is the digit itself (never changes). Place value = digit × position value. In 5,738, the face value of 7 is 7, but its place value is 700.

• **Indian Place Value System**: Uses periods—Units (ones, tens, hundreds), Thousands (thousands, ten-thousands), Lakhs (lakhs, ten-lakhs), Crores. Commas placed after 3 digits from right, then every 2 digits.

• **Factors**: Numbers that divide a given number exactly (remainder = 0). Every number has at least two factors: 1 and itself. Example: Factors of 12 = {1, 2, 3, 4, 6, 12}.

• **Multiples**: Numbers obtained by multiplying a given number by natural numbers. Multiples of 5 = {5, 10, 15, 20, ...}. A number has infinite multiples but finite factors.

• **Prime Numbers**: Numbers greater than 1 with exactly two factors (1 and itself). First ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note: 2 is the only even prime.

• **Composite Numbers**: Numbers greater than 1 with more than two factors. Example: 4, 6, 8, 9, 10. Note: 1 is neither prime nor composite.

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Formulas / Key Facts

| Concept | Formula / Rule | |---------|----------------| | Place value | Digit × Position value (ones = 1, tens = 10, hundreds = 100, ...) | | Expanded form | 4,352 = 4×1000 + 3×100 + 5×10 + 2×1 | | Number of factors | If N = p^a × q^b × r^c, then total factors = (a+1)(b+1)(c+1) | | Sum of first n natural numbers | n(n+1)/2 | | Sum of first n whole numbers | Same as above (since 0 adds nothing) | | Product of two numbers | LCM × HCF = Product of the two numbers | | Divisibility by 2 | Last digit is 0, 2, 4, 6 or 8 | | Divisibility by 3 | Sum of digits divisible by 3 | | Divisibility by 5 | Last digit is 0 or 5 | | Divisibility by 9 | Sum of digits divisible by 9 | | Divisibility by 11 | Difference of sum of alternate digits is 0 or divisible by 11 |