Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Number Systems ↓


LCM and HCF


When learning about numbers in math, especially in the context of divisibility and factors, two essential concepts are the least common multiple (LCM) and the greatest common factor (HCF), also known as the greatest common factor (GCD). These two ideas are fundamental in solving many math problems, including fractions, ratios, and algebraic expressions.

What is LCM?

The least common multiple of two or more numbers is the smallest number that is a multiple of each of the given numbers. To better understand LCM, it is helpful to first think about ‘multiples’. The multiple of a number is what you get when you multiply that number by an integer. For example, the multiples of 3 are 3, 6, 9, 12, 15, ...

Finding the LCM: A step-by-step guide

Let’s find the LCM using a simple example with the numbers 4 and 5.

Step 1: List the multiples

Number 4: 4, 8, 12, 16, 20, 24, 28, ...
Number 5: 5, 10, 15, 20, 25, 30, ...

Note that the number 20 appears in both lists. It is the first (or smallest) common number in both lists of multiples, making it the LCM.

Step 2: Use prime factorization

Another method of finding the LCM is through prime factorization:

  • Find the prime factors of each number.
  • Multiply each factor by the maximum number of times it occurs in any number.
    •     4 = 2 * 2 = 2²
    5 = 5 = 5¹
    LCM = 2² * 5¹ = 4 * 5 = 20

What is HCF?

The greatest common factor of two or more numbers is the largest number that divides each number without leaving a remainder. We are interested in the largest number that both or all of the numbers share as a factor.

Finding the HCF: A step-by-step guide

Let us find the HCF of 18 and 24.

Step 1: List the factors

Number 18: 1, 2, 3, 6, 9, 18
Number 24: 1, 2, 3, 4, 6, 8, 12, 24

Look at the largest number that appears in both lists. Here, the number 6 appears in both lists and is the largest, so the HCF is 6.

Step 2: Use prime factorization

Like the LCM, the HCF can also be found using prime factorization:

  • Find the prime factors of each number.
  • Multiply the common factors occurring in all the numbers.
    •     18 = 2 * 3²
    24 = 2³ * 3
    Lowest power common factors: 2¹ * 3¹ = 6

Visual example: LCM and HCF together

Number 1 : 15 (3 * 5) Number 2 : 20 (2² * 5) LCM: 2² * 3 * 5 = 60 HCF: 5

Applications of LCM and HCF

Understanding how to find the LCM and HCF of numbers can be incredibly useful in various areas of mathematics and its applications:

  • Reducing Fractions: In problems with fractions, the HCF can be used to simplify or reduce fractions to their lowest terms.
  • Addition of Fractions: When adding or subtracting fractions, finding the LCM of the denominators helps in finding the common denominator.
  • Problem Solving: LCM helps in solving word problems involving events that occur repeatedly at regular intervals.
  • Financial Calculations: It is used in problems related to calculation of time intervals, percentages or similar financial calculations.

Example problems

Example problem 1: LCM of 12 and 15

Uses of prime factorization:

12 = 2² * 3
15 = 3 * 5
LCM = 2² * 3 * 5 = 60

Thus, the LCM of 12 and 15 is 60.

Example problem 2: HCF of 8 and 12

Uses of prime factorization:

8 = 2³
12 = 2² * 3
HCF = 2² = 4

Thus, the HCF of 8 and 12 is 4.

Conclusion

It is essential for students to understand and be able to calculate both the LCM and HCF of numbers. These concepts are an integral part of the mathematical curriculum and are tools that make it easier to understand and apply many advanced topics in mathematics. By becoming familiar with these operations, students can enhance their problem-solving skills and mathematical reasoning.


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LCM and HCF

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