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 ↓


Euclid's Division Lemma


Euclid's division lemma is a fundamental concept in number theory and forms the basis of many other mathematical calculations. It is a straightforward way of expressing the division process, which divides something into parts. The concept is named after the Greek mathematician Euclid, who introduced it in his book "Elements". Although the idea was formulated over 2000 years ago, it is still used in mathematics today. Let's understand this concept in depth through various explanations and examples.

Understanding the lemma

Euclid's division lemma states that for any given two positive integers, a and b, there exist unique integers q and r such that:

a = bq + r

Here:

  • a is the dividend.
  • b is the divisor.
  • q is the quotient.
  • r remains.

The remainder, r, must satisfy the following condition:

0 ≤ r < b

Importance of the lemma

Euclid's division lemma is essential because it formalizes the division process, which is the basis for further concepts such as the Euclidean algorithm for finding the greatest common divisor (GCD). Using the lemma, we can also write down proofs of the existence and uniqueness of the quotient and remainder in a division, confirming that the division has been performed correctly.

Visual example

Let's try to look at the division of two integers using this lemma. Consider dividing 13 by 4:

13 (Dividends) 4 r = 1 q = 3

In this view, the green box represents the divisor 4 which is repeated 3 times within the dividend 13 (because q = 3). The 1 is unboxed, indicating the remainder r = 1.

Step-by-step example

Consider dividing 27 by 5 using Euclid's division lemma. We begin by determining how many times 5 fits into 27 without exceeding it, which will give us the quotient.

  1. 27 ÷ 5 = 5 times, some number remains (remainder).
  2. When 5 is multiplied by 5, we get 5 × 5 = 25.
  3. Subtracting 25 from 27 gives the remainder: r = 27 - 25 = 2.

Using Euclid's division lemma, we write:

27 = 5 × 5 + 2

Here, the quotient is q = 5 and remainder is r = 2, which satisfies 0 ≤ r < 5.

Why is the balance important?

The remainder provides essential information. It helps to understand how much is left after division and is important for applications such as finding the greatest common divisor (GCD) of two numbers using the Euclidean algorithm. If the remainder is zero, it indicates that one number is a perfect multiple of the other number.

Another example: division by zero and unit

Let's explore other cases using the division lemma. Consider dividing a number by 1 and by itself:

  • Dividing by 1: Choose a = 15 and b = 1. 
    15 = 1 × 15 + 0
    Here the quotient is 15 and the remainder is 0.
  • Dividing a number by itself: Choose a = 9 and b = 9. 
    9 = 9 × 1 + 0
    The quotient is 1 and the remainder is 0.

Using the lemma to find the GCD

The Euclidean algorithm, which finds the greatest common divisor (GCD) of two numbers, is based on Euclid's division lemma. Below is the procedure of the algorithm using the lemma:

Step-by-step process for finding GCD

Consider two numbers a and b:

  1. Keep using the division lemma until r = 0.
  2. Change the position of the numbers each time. a becomes b, and b becomes r.
  3. When r becomes zero then the number which comes in place of b is GCD.

Let us illustrate this with an example of finding the GCD of 56 and 98:

  1. a = 98, b = 56
  2. Divide 98 by 56: 98 = 56 × 1 + 42
  3. Substitute: a = 56, b = 42
  4. Divide 56 by 42: 56 = 42 × 1 + 14
  5. Substitute: a = 42, b = 14
  6. Divide 42 by 14: 42 = 14 × 3 + 0

Now the remainder is 0, and the GCD is 14.

Conclusion

Euclid's division lemma is a simple but powerful tool in mathematics. It lays the groundwork for understanding the division process and leads to general methods for solving mathematical problems, such as finding the GCD. Its simplicity makes it accessible, while its usefulness in calculations makes it foundational within the field.


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Euclid's Division Lemma

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