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
**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 of sum of digits at odd places and even places is 0 or divisible by 11
**HCF and LCM:**
HCF × LCM = Product of two numbers
HCF of co-prime numbers = 1
LCM of co-prime numbers = Product of the numbers
HCF of a number with its multiple = the smaller number
LCM of a number with its multiple = the larger number
Worked Examples
**Example 1: Divisibility Test** *Is 2,574 divisible by 6?*
Step 1: Check divisibility by 2 → Last digit is 4 (even) ✓ Step 2: Check divisibility by 3 → Sum of digits = 2 + 5 + 7 + 4 = 18 Step 3: Is 18 divisible by 3? → 18 ÷ 3 = 6 ✓ Since divisible by both 2 and 3, **2,574 is divisible by 6**.
**Example 2: Finding HCF by Prime Factorisation** *Find HCF of 48 and 72.*
Step 1: Prime factorisation of 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3¹ Step 2: Prime factorisation of 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3² Step 3: HCF = Product of common primes with lowest powers = 2³ × 3¹ = 8 × 3 = **24**
**Example 3: Finding LCM** *Find LCM of 12, 15, and 20.*
Step 1: Prime factorisations:
12 = 2² × 3
15 = 3 × 5
20 = 2² × 5
Step 2: LCM = Product of all primes with highest powers = 2² × 3 × 5 = 4 × 3 × 5 = **60**
**Example 4: Using HCF-LCM Relationship** *The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.*
Using: HCF × LCM = Product of two numbers 12 × 180 = 36 × Other number 2160 = 36 × Other number Other number = 2160 ÷ 36 = **60**
Common Mistakes
**Confusing place value with face value** → Place value depends on position (7 in 472 has place value 70), while face value is the digit itself (7). Always identify what the question asks.
**Applying divisibility rule for 4 to the entire number** → Only the last two digits matter for divisibility by 4. For 3,128, check if 28 is divisible by 4 (yes, 28 ÷ 4 = 7), not the whole number.
**Taking highest powers for HCF instead of lowest** → HCF uses common factors with lowest powers; LCM uses all factors with highest powers. Reverse this and you'll swap answers.
**Forgetting that HCF × LCM = a × b only works for two numbers** → This relationship does not extend directly to three or more numbers. For three numbers, use the formula involving pairwise HCFs.
**Assuming all integers are whole numbers** → Negative integers exist. When a question asks "how many integers between -5 and 5," include negatives and possibly zero depending on whether endpoints are included.
A number when divided by 12 leaves a remainder of 7. What will be the remainder when the same number is divided by 6?
Q2 · Numbers · MEDIUM
The HCF of two numbers is 18 and their LCM is 360. If one of the numbers is 72, what is the other number?
Q3 · Numbers · MEDIUM
What is the smallest number that should be added to 2957 so that the resulting number is divisible by 11?
Q4 · Numbers · MEDIUM
The sum of the digits of a two-digit number is 12. If the digits are reversed, the new number is 18 less than the original number. What is the original number?
Q5 · Numbers · HARD
Three bells ring at intervals of 12 minutes, 15 minutes and 18 minutes respectively. If they all ring together at 8:00 AM, at what time will they next ring together?