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 ↓


Prime Factorization


Prime factorization is the process of expressing a number as a product of its prime factors. This concept is a fundamental idea within number theory and has applications in a variety of fields, including cryptography, mathematics, and computer science.

Before we move further into prime factorization, it is important to understand what prime numbers are. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, a prime number has exactly two distinct positive divisors: 1 and itself.

Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, etc. Note that these numbers have no divisors other than 1 and themselves.

Basics of prime factorization

The goal of prime factorization is to break down a composite number (a number that is not prime) into a product of prime numbers. For example, the prime factorization of 12 can be determined by successively dividing it by prime numbers.

Example

Let's do prime factorization on the number 12.

  • Step 1: 12 is an even number, so we can start by dividing it by the smallest prime number 2.
    12 ÷ 2 = 6
  • Step 2: 6 is also an even number, so divide it by 2 again.
    6 ÷ 2 = 3
  • Step 3: 3 is a prime number, so we stop here.

The prime factorization of 12 is:

2 x 2 x 3

We can also write this using exponents:

2 2 x 3 1

12 2 6 2 3

Why prime factorization?

Prime factorization has several important applications:

  • Greatest Common Divisor (GCD): To find the GCD of two numbers, we factor each number, then take the smallest exponent for each common prime factor.
  • Least Common Multiple (LCM): Finding the LCM involves taking the highest exponent of each prime factor of the numbers under consideration.
  • Coding theory: Prime factorization plays a vital role in encryption systems, such as RSA encryption, making it fundamental in secure communications.

How to do prime factorization

To get started with prime factorization, you follow these steps:

  1. Start with the smallest prime number, which is 2.
  2. Check if 2 divides the number exactly (i.e., without leaving a remainder). If it does, divide the number by 2.
  3. Continue dividing by 2 until there is no number left.
  4. Move to the next smallest prime number, which is 3, and repeat the steps.
  5. Continue this process with successive prime numbers (5, 7, 11, 13, ...) until the quotient is a prime number.

Detailed example

Let's take a detailed step-by-step factorization of 60.

  • Step 1: Start at 2 (because 60 is even).
    60 ÷ 2 = 30
  • Step 2: Divide 30 by 2 (this is still an even number).
    30 ÷ 2 = 15
  • Step 3: Move on to the next prime number, which is 3 (15 is not divisible by 2).
    15 ÷ 3 = 5
  • Step 4: 5 is already a prime number. We end our factorization here.

The prime factorization of 60 is:

2 x 2 x 3 x 5

60 2 30 2 15 3 5

Factorization of large numbers

Factoring small numbers is simple, but what about larger numbers? The process is the same, but it may require checking divisibility by more prime numbers. Let's practice with an example involving a larger number, 126.

Example: Prime factorization of 126

  • Start with Step 1: 2.
    126 ÷ 2 = 63
  • Step 2: Go to 3 (63 is odd).
    63 ÷ 3 = 21
  • Step 3: Divide 21 by 3.
    21 ÷ 3 = 7
  • Step 4: 7 is a prime number; factorization is complete.
    7

The prime factorization of 126 is:

2 x 3 x 3 x 7

Using exponents this can be written as:

2 1 x 3 2 x 7 1

126 2 63 3 21 3 7

Applications of prime factorization

Understanding prime factorization of numbers has many beneficial applications:

Finding the greatest common divisor (GCD)

Suppose you want to find the GCD of 48 and 180. First, perform prime factorization on each number.

  • 48:
    48 ÷ 2 = 24

    24 ÷ 2 = 12

    12 ÷ 2 = 6

    6 ÷ 2 = 3

    so, 
    48 = 2 4 x 3 1
  • 180:
    180 ÷ 2 = 90

    90 ÷ 2 = 45

    45 ÷ 3 = 15

    15 ÷ 3 = 5

    so, 
    180 = 2 2 x 3 2 x 5 1

The GCD is determined by taking the minimum degree of each common factor, hence:

GCD(48, 180) = 2 2 x 3 1 = 12

Finding the least common multiple (LCM)

Using the same factorization of 48 and 180, the LCM is obtained by taking the highest power of all prime factors:

LCM(48, 180) = 2 4 x 3 2 x 5 1 = 720

Prime factorization: cautionary note

Although prime factorization is a powerful tool, it's important to use it with an understanding of its properties:

  • Factorization is only defined for positive integers greater than 1.
  • Prime factorization is unique for each number, except for the order of the factors (called the fundamental theorem of arithmetic).
  • This may not always be easy for large numbers due to computational constraints.

Additional examples

Let us try prime factorization with some more numbers for more clarity.

Example 1: Prime factorization of 84

  • Step 1: Divide by 2.
    84 ÷ 2 = 42
  • Step 2: Divide by 2 again.
    42 ÷ 2 = 21
  • Step 3: Divide by 3.
    21 ÷ 3 = 7
  • Step 4: 7 is a prime number, stop here.

The prime factorization of 84 is:

2 2 x 3 1 x 7 1

Example 2: Prime factorization of 100

  • Step 1: Divide by 2.
    100 ÷ 2 = 50
  • Step 2: Divide by 2 again.
    50 ÷ 2 = 25
  • Step 3: Divide by 5.
    25 ÷ 5 = 5
  • Step 4: 5 is a prime number, stop here.

The prime factorization of 100 is:

2 2 x 5 2

Conclusion

Prime factorization is a powerful technique for breaking down numbers into their most basic components - the prime numbers. It not only helps simplify various mathematical calculations, such as finding the GCD and LCM, but also helps us understand deeper principles of number theory. It also plays a vital role in understanding prime factorization. Mastering prime factorization opens the door to many mathematical possibilities and problem-solving strategies. By practicing with a variety of numbers and considering its practical applications, students and enthusiasts can gain an understanding of this essential mathematical concept. can gain a complete understanding.


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Prime Factorization

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