11 and 13 are the prime factors of 143.

11, 13 are prime numbers and if we multiply them, we’ll get 143. So, 11 and 13 are the prime factors of 143.

Prime Factors | 11 and 13 |

Product of Prime Factor | 11 x 13 |

Exponential Form | 11^{1} x 13^{1} |

Total Number of Factors | 2 |

Largest Prime Factor | 13 |

Smallest Prime Factor | 11 |

Closest Prime Numbers | 141 |

## Introduction to Prime Factors

Before learning the methods of calculating prime factors, we should get familiar with the term prime factors. What are prime factors? And why are they called prime factors? Prime factors are those integers which can express a given number as a form of their multiplication. In other words, if we multiply two prime numbers, we’ll get another number and those previous numbers are the prime factors of the later one. We’ll be more clear with an example :

105 = 3 x 5 x 7

Here, the multiplication of the prime numbers 3, 5 & 7 form 105. So, 3, 5 & 7 are the prime factors of 105. From the above example, we get to know another important thing: our given number is always evenly divisible by the prime factors. As we see, 105 is evenly divisible by 3, 5 & 7. They’re being called prime factors because they are prime numbers. If they weren't prime numbers, they would’ve been called only factors.

## Definition of Factorization

Factorization is the process to determine all the numbers that exactly divide a given number. We can find all those numbers through different calculation methods. But remember, the given number as well as the factors must be integer numbers.

## Definition of Prime Factorization

Prime factorization is also a kind of factorization but the only difference is the factors are prime numbers in this case.

## Prime Factor's Formula

A prime factor must be a prime number as well as a factor of the given number. Basically, prime factors can be found by decomposing our given number. It can be expressed as a product of prime numbers with orders. In general, we represent our given number as a product of prime numbers with their orders. These prime numbers are certainly the prime factors of the given number.

N = p_{f1}^{a1} + p_{f2}^{a2} + p_{f3}^{a3} + ...... + p_{fn}^{an}

N = Any integer number

p_{f1}, p_{f2}, p_{f3}, p_{fn} = Prime factors

a_{1}, a_{2}, a_{3}, a_{n} = Orders of prime factors

## How to Calculate the Prime Factors of 143?

Numerous methods exist for identifying prime factors. Most often used techniques include:

- Factor Tree Method.
- Division Method.

### Factor Tree Method

In factor tree method, the given number and the factors of the given number will be connected via diagonals like this:

Hence, it is called the Factor Tree Method. Here we’ll draw the given number as a root & factors as a tree. From the tree we’ll filter those factors which are prime numbers. We can do this step by step.

#### Step 1

Let's use the given number, 143. We'll now start by writing the root. Then, we'll draw two arrows to connect the first to the root's branches or contributing elements. Starting with 2, we'll keep trying until we find a number that divides 143 perfectly. (Note that we didn't take 1 because every number can be divided by both 1 and the number itself).

Without a remainder, 143 cannot be divided by 2. Since 143 is easily divisible by 11, we'll give it a go with 11 (Note: we would have to continue exploring until we found a prime number that exactly divided 143). So, as the first two factors of 143, we'll obtain 11 and 13.

** Check the first step of these prime factorization examples to better understand how this step is done:**

- Prime factorization of 142
- Prime factorization of 141
- Prime factorization of 140

#### Step 2

11 & 13 are prime numbers. So, the process ends here & we got our prime factors. Else, we had to continue the process until every branch of the tree ultimately ended as a prime number.

From the diagram we get 11, 13 as the prime factors of 143.

We can express like this: 143 = 11 x 13

**Note: we need to factorize them until all the factors become prime numbers.**

### Division Method

We will now discuss the division method. We can guess that this approach is connected to division operation from the name. This technique is actually quite easy to use. Simply keep dividing the given number until the quotient equals one.

Let's go over the specifics of this procedure using the given number 143.

#### Step 1

Let’s implement the division method on 143 to find out its prime factors. First we’ll try to divide the 143 by 2 as 2 is the smallest prime number. But 2 can’t divide 143 exactly. So, we’ll try with 11 this time as 11 is the next prime number which can divide 143 evenly giving 13 as the quotient.

**Check the first step of these prime factorization examples to better understand how this step is done:**

- Prime factorization of 144
- Prime factorization of 145
- Prime factorization of 146

#### Step 2

As 13 is a prime number, the smallest & only prime number to divide 13 exactly is 13 itself and leaves 1 as quotient. So, we found the quotient 1 hence the division method is done.

We got 11, 13 as the prime factors of 143.**Note: We’ll have to repeat the process until the quotient becomes 1.**

## Non-Prime Factors of 143

All the positive factors of 143 are 1, 11, 13, 143. So, the non-prime factors are 1, 143

## Negative Factors of 143

The negative factors of 143 are -1, -11, -13, -143

## How to Determine All the Factors of 143?

To determine all the factors of 143, we have to find every divisor that divides 143 exactly. After finding that, we should express this like this:

143 ÷ 1 = 143

143 ÷ 11 = 13

Here every divisor & quotient are the factors of 143.

So, the positive factors of 10365 are: 1, 11, 13, 143.

We can also express this like:

-143 x -1 = 143

-13 x -11 = 143

So the negative factors are: -1, -11, -13, -143.

Remember, a negative factor must multiply with another negative factor only to get our given number.

## Facts of Factorization

- Factors can’t be a fragment of a number.
- Given number must be an integer.
- Factors can be both negative & positive.
- 1 is the factor of every natural number.
- A quadratic equation can also have factors.
- If we divide a given number, then the divisors & the quotient of the given number are also factors of it. Example:

143 ÷ 1 = 143

143 ÷ 11 = 13

Here, 1, 11, 13, 143 are the factors of 143.

## Facts About Factorization

We use factors to arrange something in different ways. It helps us to divide something equally. In mathematics, it is used in various number related problems. Besides, it is used in comparing something, money exchange, understanding time etc. We can also factorize quadratic equations to simplify it.

## Frequently Asked Questions

### 1. Can Factors Be Negative?

Yes. Factors can be negative too. Like the factors of 10 are 1, 2, 5, 10, -1, -2, -5, -10. Because if we multiply -10 with -1, we’ll get 10. So, -10 & -1 are the factors of 10. But most of the time we use positive factors only.

### 2. Is 143 a Square Number?

No. The square root of 10365 isn’t an integer. So it isn’t a square number.

### 3. What Is the Square of 143?

Square of 143 is 20449.

### 4. What Is the Root of 143?

Root of 143 is 11.958260743101398

### 5. Is 143 a Composite Number or a Prime Number?

143 is a composite number.

### 6. How Many Factors Does a Prime Number Have?

A Prime number has only 2 factors. They are 1 & the number itself.

### 7. What is a Composite Number?

If a positive integer number has more than two factors, it can be called a composite number.

### 8. What are the factors of a prime number?

They are 1 & the number itself.

S. M. Fahim

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