Number System
TN TET Mathematics
Overview
The Number System forms the backbone of primary mathematics and is a foundational topic for TN TET Paper I and Paper II. Questions from this topic test your understanding of whole numbers, integers, place value, divisibility rules, and finding HCF and LCM.
Mastery here means quick mental calculations during the exam. You must be fluent with divisibility tests (saves time on HCF/LCM problems), comfortable moving between number types, and confident with place value manipulation. Pedagogy questions may also ask how to teach these concepts to children—connecting abstract numbers to concrete understanding.
Key Concepts
- Natural Numbers (N): Counting numbers starting from 1. {1, 2, 3, 4, ...}. Zero is NOT included.
- Whole Numbers (W): Natural numbers plus zero. {0, 1, 2, 3, ...}. Every natural number is a whole number.
- Integers (Z): Whole numbers plus negative numbers. {..., -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. Place value = digit × position value. In 4,725: the face value of 7 is 7, but its place value is 700.
- Divisibility: A number is divisible by another if the remainder is 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. 12's factors: 1, 2, 3, 4, 6, 12. 12's multiples: 12, 24, 36...
- HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Also called GCD.
- LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers.
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 0, 2, 4, 6, or 8
- By 3: Sum of 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 digits is divisible by 9
- By 10: Last digit is 0
- By 11: Difference between sum of digits at odd places and even places is 0 or divisible by 11
HCF and LCM Relationship
- HCF × LCM = Product of the two numbers
- For numbers a and b: HCF(a, b) × LCM(a, b) = a × b
Properties of HCF and LCM
- HCF of co-prime numbers = 1
- LCM of co-prime numbers = their product
- HCF ≤ smaller number
- LCM ≥ larger number
Worked Examples
Example 1: Place Value Find the difference between the place value and face value of 6 in 46,823.
Solution:
- Face value of 6 = 6
- Place value of 6 = 6 × 1,000 = 6,000
- Difference = 6,000 − 6 = 5,994
Example 2: Divisibility Test Is 2,574 divisible by 6?
Solution: For divisibility by 6, check both 2 and 3:
- By 2: Last digit is 4 (even) ✓
- By 3: Sum of digits = 2 + 5 + 7 + 4 = 18. Is 18 ÷ 3 = 6? ✓
Since divisible by both 2 and 3, yes, 2,574 is divisible by 6.
Example 3: HCF using Prime Factorisation Find HCF of 36 and 48.
Solution:
- 36 = 2² × 3²
- 48 = 2⁴ × 3¹
HCF = Product of common prime factors with lowest powers
- Common factors: 2 and 3
- Lowest powers: 2² and 3¹
HCF = 2² × 3 = 4 × 3 = 12
Example 4: LCM using Prime Factorisation Find LCM of 15 and 20.
Solution:
- 15 = 3 × 5
- 20 = 2² × 5
LCM = Product of all prime factors with highest powers
- Factors: 2², 3, 5
LCM = 4 × 3 × 5 = 60
Example 5: Using HCF × LCM = Product HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.
Solution:
- HCF × LCM = Product of numbers
- 12 × 180 = 36 × other number
- 2,160 = 36 × other number
- Other number = 2,160 ÷ 36 = 60
Common Mistakes
- Confusing place value with face value → Remember: face value never changes; place value depends on position. The 5 in 5,000 has face value 5 but place value 5,000.
- Forgetting zero in divisibility by 10 → Students remember "ends in 0 or 5" for divisibility by 5 and wrongly apply it to 10. For 10, the number must end in 0 only.
- Adding instead of multiplying for LCM → LCM is found by multiplying prime factors with highest powers, not by adding numbers. Always use factorisation or division method.
- Mixing up HCF and LCM methods → For HCF, take lowest powers of common factors. For LCM, take highest powers of all factors. A mnemonic: HCF = "Humble" (smaller), LCM = "Larger" (bigger result).
- Applying divisibility rule for 4 to single digit → The rule for 4 requires checking the last two digits, not just the last digit. 312 is divisible by 4 because 12 ÷ 4 = 3, not because 2 is even.
- Assuming HCF × LCM formula works for three numbers → The formula HCF × LCM = product works only for two numbers. For three or more numbers, use factorisation directly.
Quick Reference
- Number hierarchy: Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real
- Divisibility by 6 = check 2 AND 3 together
- HCF × LCM = a × b (for two numbers only)
- Co-prime numbers: HCF = 1, LCM = product
- Divisibility by 11: (Sum of odd-place digits) − (Sum of even-place digits) = 0 or multiple of 11
- Quick LCM shortcut: For two numbers, LCM = (a × b) ÷ HCF