## What Is a Factor?

A factor is a number that divides another number, leaving no remainder. In other words, if multiplying two whole numbers gives us a product, then the numbers we are multiplying are factors of the product because they are divisible by the product.

There are two methods of finding factors: multiplication and division. In addition, rules of divisibility may also be used.

**Example: **Let us consider the number 8. 8 can be a product of 1 and 8, and 2 and 4. As a result, the factors of 8 are 1, 2, 4, 8. Therefore, when finding or solving problems involving factors, only positive numbers, whole numbers, and non-fractional numbers are considered.

A general formula to remember is that a and b are factors of ab’s product.

2 ✕ 3 = 6. Therefore, 2 and 3 are factors of 6. There is no remainder when 6 is divided by either 2 or 3.

9 ✕ 3 = 27. Therefore, 9 and 3 are factors of 27. There is no remainder when 27 is divided by either 9 or 3.

7 ✕ 5 = 35. Therefore, 5 and 7 are factors of 35. There is no remainder when 35 is divided by either 5 or 7.

**Example:** Find all the factors of the number 10.

Properties of Factors

- All integers have a finite number of factors.
- A number’s factor is always less than or equal to the number; it can never be bigger than the number.
- Except for 0 and 1, every integer has a minimum of two factors: 1 and the number itself.
- Factors are found by employing division and multiplication.
**Fun Facts about Factors**

- Factors are never decimals or fractions; they are only whole numbers or integers.
- All even numbers have 2 as a factor.
- 5 is a factor for all numbers that end in 0 and 5.
- All numbers higher than 0 and ending in a 0 have 2, 5, and 10 as factors.
- Factoring is a common way to solve or simplify algebraic expressions.

## Prime Factorization

When we write a number as a product of all its prime factors, it is called prime factorization. Every number in prime factorization is a prime number. To write the number as a product of prime factors, sometimes we might have to repeat the factors too.

**Example:** To write the prime factorization of 8, we can write 8 = 2 ✕ 2 ✕ 2, i.e, the prime factor 2 is repeated three times.

Real-life Applications of Factorization

**Equal division.** If six people come together to eat a whole pizza that has been cut into 24 slices, it would only be fair that everyone receives an equal number of slices. Therefore, this pizza can be divided into equal shares because 6 (the number of people) is a factor of 24 (the number of pizza slices). When you divide 24 by 6, you get 4, and each individual receives four slices!

**Factoring and money.** The exchange of money and its divisions into smaller units rely heavily on factoring. For example, four quarters equal one dollar in America. In India, a rupee was further divided into 1 paisa, 5 paise, 10 paise, 25 paise, and 50 paise.

## Solved Examples

**Example 1: **Find all the factors of 20.

Step** 1:** Write all the numbers from 1 to 20.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

**Step 2**: Now check which of these numbers are divisible by 20 and leave no remainders.

20/1 = 20

20/2 = 10

20/3 = not divisible.

Continue dividing 20 by each of these numbers.

S**tep 3**:** **The factors of 20 are 1,2,4,5, 10, and 20.

**Example 2:** Find all the factors of 31.

31 is a prime number. The only two numbers that divide 31 completely are 1 and 31.

Therefore, factors of 31 are 1 and 31.

**Example 3:** Find the prime factors of 144.

Just as the name says, prime factorization is the method of deriving the prime factors of any number. Prime factors are prime numbers. The factors of such numbers are 1 and the number itself. For example, 13 is a prime number because the factors of this number are 1 and 13.

Consider the number 144. Start by considering the smallest possible factor, i.e., 2.

144 = 2 x 72 = 2 x 2 x 36 = 2 x 2 x 2 x 18 = 2 x 2 x 2 x 2 x 9 = 2 x 2 x 2 x 2 x 3 x 3

Thus, the prime factors of 144 are 2 and 3 as these factors are prime numbers.

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## Practice Problems

## Factor

### 1Which of the following option represents all the factors of the number 10.

All other options have 3 as one of the factors whereas 3 does not divide number 10 completely. When 10 is divided by 3, we get a remainder.

### 2Which of the following option represents all factors of the number 27.

Option 2 and 3 have number four (4) as one of the factors of 27 whereas number 4 does not divide 27 completely. Option 4 has number 5 as one of the factors of 27 which is incorrect as 5 does not divide 27 completely. When 27 is divided by 4 or 5 we get a remainder, hence 4 and 5 are not factors of 27.

### 3Which of the following option represents all the factors of the given number 12.

There is no remainder left when 12 is divided by 1, 3, 4, 2, 6 and 12.

### 4Select the option with all the factors of the number 15

There is no remainder left when 15 is divided by 1, 3, 5 and 15.

## Frequently Asked Questions

**What are the factors?**

A factor is a number that can be multiplied to create a specific number in math (for example, 5 and 8 are factors of 40).

**How is factoring used in real life?**

In real life, factoring is a valuable skill. Typical applications include dividing something into equal portions, exchanging money, comparing prices, grasping time, and making computations while traveling.

**What is the importance of learning about prime factors?**

Prime factors are significant for people who try to build (or break) secret codes based on numbers and need to know about factorization. It is known as cryptography or encryption. Due to the difficulty of factoring very big numbers, which can take a long time for computers to do.

**How do you find the factor in math?**

Here is a step-by-step guide to finding the factor of any number in math. 1. Start by considering the smallest natural numbers like 2, 3, etc.

2. Divide the number by the smallest natural number that it could be divided by.

3. Continue dividing the number by the smallest possible natural number, which gives 0 as the remainder.

4. Stop when the number is wholly divided to give 1 as the quotient.

5. The numbers that you used for dividing the original number are the factors of the number.