In mathematics, factorization is the process of breaking down a number into its factors, which are the numbers that can divide the original number evenly without leaving a remainder. Factorization can be useful in many applications, including simplifying fractions, finding the greatest common factor, and solving equations. In this article, we will discuss how to factorize 24.

 

To factorize 24, we need to find the factors of 24. We can start by listing the numbers that divide 24 evenly. These numbers are 1, 2, 3, 4, 6, 8, 12, and 24. Therefore, 1, 2, 3, 4, 6, 8, 12, and 24 are the factors of 24.

 

i.e. F₂₄ = {1, 2, 3, 4, 6, 8, 12, 24}

 

In mathematics, a factor is a number that can divide another number evenly without leaving a remainder. The factors of a number can be useful in various applications, including prime factorization, finding common denominators, and solving equations. In this article, we will discuss the factors of 75.

 

To find the factors of 75, we can start by listing the numbers that divide 75 evenly. These numbers are 1, 3, 5, 15, 25, and 75. Therefore, 1, 3, 5, 15, 25, and 75 are the factors of 75.

 

i.e. F₇₅ = {1, 3, 5, 15, 25, 75}

 

The series associated with a geometric sequence is known as a geometric series. For example, 2 + 4 + 8 + 16 + 32 is the geometric series associated with the geometric sequence 2, 4, 8, 16, 32.

 

If three numbers a, b and c are in geometric progression GP, then the middle term b is known as the geometric mean (GM) between the other two. For example, 24, 12 and 6 are in GP, so 12 is the geometric mean between 24 and 6.

Consider the following sequences.

 

a. 2, 4, 8, 16, … … … …

b. 27, 9, 3, 1, … … … …

c. 16, -24, 36, -54, … … … …

 

In the first sequence, each term is double of its previous term. In the second sequence, each term is one-third of its previous term. Each term of the third sequence is -3/2 times its previous term. In each of the above sequence, the ratio of a term and its previous term is equal or constant. Such a sequence is known as a Geometric Sequence or Geometric Progression.

Arithmetic Series

 

The series associated with an arithmetic sequence is known as an arithmetic series. For example: 3 + 7 + 11 + 15 + 19 + 23 + 27 + 31 is the arithmetic series associated with the arithmetic sequence: 3, 7, 11, 15, 19, 23, 27, 31.

 

If three numbers a, b and c are in Arithmetic Sequence (AS) then the middle term b is said to be the arithmetic mean (AM) between the other two. For example, 10, 14 and 18 are in AS, so 14 is the arithmetic mean between 10 and 18. If any number of terms are in AS then the terms between the first and the last terms are known as the arithmetic means between the first and the last. For example, 11, 14, 17, 20, 23, 26, 29, 32 are in AS, so 14, 17, 20, 23, 26, 29 are the arithmetic means between 11 and 32.

 

Consider the following sequences.

 

a. 1, 6, 11, 16, ... … …

b. 6, 2, -2, -6, … … … …

c. 3, 4.5, 6, 7.5, … … …

 

In the first sequence, each term is increased by 5 than the preceding term. In the second sequence, each term is decreased by 4 than the preceding term. Each term in the third sequence is increased by 1.5 than the preceding term. In each of the above sequences, the difference between a term and its preceding term is equal or constant. Such a sequence is said to be an Arithmetic Sequence or Arithmetic Progression.

 

Let A = {2, 3, 4}, B = {6, 9, 12} and C = {a, b, c}. If f : A → B and g : B → C are defined by f = {(2, 6), (3, 9), (4, 12)} and g = {(6, a), (9, b), (12, c)},

f is the function from A to B such that

 

2 ∈ A, f(2) = 6 ∈ B

3 ∈ A, f(3) = 9 ∈ B

4 ∈ A, f(4) = 12 ∈ B

 

Range of f = {6, 9, 12} = domain of g.