Topics
Number System
Number System(Consolidating the Sense of Numberness)
Number System (Including Estimation and Approximation)
Theme 1 : Numbers
Estimation
Numbers in Indian and International Systems (With Comparison)
Theme 2 : Ratio, Proportion and Arithmetic Problems
Ratio and Proportion
Numbers in India and International System (With Comparison)
Theme 3 : Algebra
Natural Numbers and Whole Numbers (Including Number Line and Patterns)
Algebra
Negative Numbers and Integers
Place Value
Theme 4 : Geometry
Geometry
Theme 5 : Mensuration
Mensuration
Natural Numbers and Whole Numbers (Including Patterns)
Sets
Data Handling
Negative Numbers and Integers
Theme 6 : Statistics
Fractions
Number Line
Decimal Fractions
Playing with Numbers (Including H.C.F. and L.C.M.)
HCF and LCM
Ratio and Proportion (Including Unitary Method)
Playing with Numbers
- Simplification of Brackets
- Finding Factors Using Rectangular Arrangements and Division
- Factors and Common Factors
- Multiples and Common Multiples
- Concept of Even and Odd Number
- Tests for Divisibility of Numbers
- Divisibility by 2
- Divisibility by 4
- Divisibility by 8
- Divisibility by 3
- Divisibility by 6
- Divisibility by 9
- Divisibility by 5
- Divisibility by 11
Percent (Percentage)
Sets
Idea of Speed, Distance and Time
Ratio
Proportion (Including Word Problems)
Fundamental Concepts (With operations on Algebraic Expressions)
Framing Algebraic Expressions (Including Substitution and Linear Equations)
Unitary Method
Fundamental Concepts (Including Angles and their Properties)
Fractions
- Concept of Fraction
- Types of Fractions
- Concept of Proper and Improper Fractions
- Concept of Mixed Fractions
- Like and Unlike Fraction
- Concept of Equivalent Fractions
- Conversion between Improper and Mixed fraction
- Conversion between Unlike and Like Fractions
- Simplest Form of a Fractions
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of Fraction
- Division of Fractions
- Using Operator 'Of' with Multiplication and Division
- BODMAS Rule
- Problems Based on Fraction
Triangles (Including Types and Properties of Triangles)
Decimal Fractions
Percent (Percentage)
Quadrilaterals
Circles
Idea of Speed, Distance and Time
Constructions
Symmetry
Fundamental Concepts
Fundamental Operations (Related to Algebraic Expressions)
Recognition of Solids
Substitution (Including Use of Brackets as Grouping Symbols)
Framing Algebraic Expressions (Including Evaluation)
Perimeter and Area of Plane Figures
Simple (Linear) Equations (Including Word Problems)
Data Handling (Including Mean and Median)
Fundamental Concepts
Angles (With Their Types)
Properties of Angles and Lines (Including Parallel Lines)
Triangles (Including Types, Properties and Constructions)
Quadrilateral
Polygons
The Circle
Symmetry (Including Constructions on Symmetry)
Recognition of Solids
Perimeter and Area of Plane Figures
Data Handling (Including Pictograph and Bar Graph)
Mean and Median
- Definition: Prime Factor
- Examples
- Prime Factorisation of a Product
- Applications of Prime Factorisation: Checking Co-primeness
- Applications of Prime Factorization: Checking Divisibility
Definition: Prime Factor
Prime factors of a number are the prime numbers that divide it exactly.
Examples
1.24
All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Prime factors: 2 and 3
- F₂₄ = 1, 2, 3, 4, 6, 8, 12, 24
- P.F₂₄ = 2 and 3
2. 50
Factors of 50: 1, 2, 5, 10, 25, 50
Prime factors: 2 and 5
- F₅₀ = = 1, 2, 3, 4, 6, 8, 12, 24
- P.F₅₀ = 2 and 5
3. 64
Factors of 64: 1, 2, 4, 8, 16, 32, 64
Prime factor: 2
- F₆₄ = 1, 2, 4, 8, 16, 32, 64
- P.F₆₄ = 2
Prime Factorisation of a Product
When a number is written as a product of two numbers, we can find its prime factorisation by factoring each part.
Write it as a product of two factors:
72 = 12 × 6
Now factor each:
-
12 = 2 × 2 × 3
-
6 = 2 × 3
The prime factors of a product are simply the combined prime factors of its factors.
Applications of Prime Factorisation: Checking Co-primeness
Two numbers are coprime if they have no common prime factors.
Example 1: 56 and 63
Prime factorisation:
-
56 = 2 × 2 × 2 × 7
-
63 = 3 × 3 × 7
Common prime factor = 7
Not coprime
Example 2: 242 and 195
Prime factorisation:
-
242 = 2 × 11 × 11
-
195 = 3 × 5 × 13
No common prime factors
242 and 195 are coprime.
Applications of Prime Factorization: Checking Divisibility
Rule: If prime factors of B are included in the prime factors of A → A is divisible by B.
Example 1: Is 168 divisible by 12?
Prime factorisation:
-
168 = 2 × 2 × 2 × 3 × 7
- 12 = 2 × 2 × 3
Prime factors of 12 exist inside factors of 168
168 is divisible by 12
Example 2: Is 75 divisible by 21?
Prime factorisation:
-
75 = 3 × 5 × 5
-
21 = 3 × 7
Factor 7 is not present in 75
75 is NOT divisible by 21
Key Points Summary
-
Prime factors are prime numbers that divide a number exactly
-
Order doesn't matter when multiplying prime factors
-
Two numbers are co-prime if they share no common prime factors
-
A number A is divisible by B if all the prime factors of B are present in the prime factorisation of A.
Example Question 1
Factorise into primes: 117.
Vertical arrangement:
| 3 | 117 |
| 3 | 39 |
| 13 | 13 |
| 1 |
117 = 3 × 3 × 13.
Horizontal arrangement:
117 = 13 × 9
117 = 13 × 3 × 3
117 = 3 × 3 × 13.
Example Question 2
Factorise into primes: 250.
Vertical arrangement:
| 2 | 250 |
| 5 | 125 |
| 5 | 25 |
| 5 | 5 |
| 1 |
250 = 2 × 5 × 5 × 5.
Horizontal arrangement:
250 = 2 × 125
250 = 2 × 5 × 25
250 = 2 × 5 × 5 × 5.
Example Question 3
Find the prime factors of 40.
Vertical arrangement:
| 2 | 40 |
| 2 | 20 |
| 2 | 10 |
| 5 | 5 |
| 1 |
40 = 2 × 2 × 2 × 5
Horizontal arrangement:
40 = 10 × 4
40 = 5 × 2 × 2 × 2.
