Number System
Whole Numbers, Integers, Place Value, Factors and Multiples
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.
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.
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.
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 |
Must-Remember Facts:
- Smallest whole number = 0; Smallest natural number = 1
- Smallest prime number = 2; Smallest composite number = 4
- 1 is neither prime nor composite
- Zero is an even number
- Negative numbers have no concept of prime/composite at primary level
Worked Examples
Example 1: Place Value Question: In the number 83,47,562, find the place value and face value of 4.
Solution:
- Face value of 4 = 4 (the digit itself)
- Position of 4 = Ten-thousands place
- Place value of 4 = 4 × 10,000 = 40,000
Example 2: Finding All Factors Question: Find all factors of 36.
Solution:
- Start dividing from 1: 36 ÷ 1 = 36 ✓
- 36 ÷ 2 = 18 ✓
- 36 ÷ 3 = 12 ✓
- 36 ÷ 4 = 9 ✓
- 36 ÷ 6 = 6 ✓
- Factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}
- Total factors = 9
Example 3: Common Multiples Question: Find the first three common multiples of 4 and 6.
Solution:
- Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
- Multiples of 6 = 6, 12, 18, 24, 30, 36, ...
- Common multiples = 12, 24, 36, ...
- First three common multiples = 12, 24, 36
Example 4: Integer Operations Question: Solve: (−8) + 5 − (−3)
Solution:
- (−8) + 5 = −3
- −3 − (−3) = −3 + 3 = 0
- Answer = 0
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| "0 is a natural number" | Natural numbers start from 1. Zero is only a whole number. |
| "1 is a prime number because it has only one factor" | Prime numbers must have exactly TWO factors. 1 has only one factor (itself), so it is neither prime nor composite. |
| Confusing place value with face value | Face value never changes. Place value = digit × position. In 572, face value of 7 is 7, place value is 70. |
| "Factors of a number are infinite" | Factors are always finite. Multiples are infinite. 12 has only 6 factors but infinite multiples. |
| Subtracting negative: 5 − (−3) = 2 | Subtracting a negative means adding. 5 − (−3) = 5 + 3 = 8. |
| Writing Indian place value with international commas | Indian system: 45,00,000 (forty-five lakhs). International: 4,500,000. Use correct comma placement. |
Quick Reference
- Natural ⊂ Whole ⊂ Integers: Every natural is whole, every whole is integer.
- Place value formula: Digit × Position value (ones, tens, hundreds, thousands, ten-thousands, lakhs, ten-lakhs, crores).
- Factors are finite, multiples are infinite.
- 1 is neither prime nor composite; 2 is the only even prime.
- Integer rule: (+) × (−) = (−); (−) × (−) = (+).
- Divisibility shortcut: For 3 and 9, add all digits; for 11, find alternate-digit difference.