Factors of a Number

Introduction

In mathematics, a factor of a number is a divisor that divides the given number evenly, with no remainder. Factors can be identified as pairs of numbers that multiply together to make another number. It is important to note that factors are always an integer and never a fraction or decimal value. Various methods, such as the division and multiplication methods, can be employed to determine the factors of a number. Factors are essential in practical scenarios, such as dividing items into equal groups, comparing prices, and currency exchange. Let’s understand factors in detail and the process of finding them in this article.

Definition of Factor

Factors of a number are values that divide the number evenly, leaving no remainder. These factors can be either positive or negative. For instance, consider the number `12`. It's divisible by `1, 2, 3, 4, 6,` and `12`. So, the positive factors of `12` are `1, 2, 3, 4, 6,` and `12`. However, there are also negative factors: `-1, -2, -3, -4, -6,` and `-12`. It's worth noting that when you multiply two negative numbers, you get a positive result. Hence, both positive and negative factors play a role in understanding the properties of numbers, although we typically focus on the positive factors in everyday use.

Properties of Factors

Following are the properties of factors of a number:

• The number of factors for any given number is finite, meaning there's a set count of factors.
• Each factor of a number is always smaller than or equal to the given number.
• Every number except `0` and `1` has a minimum of two factors: `1` and the number itself.
• For a prime number, there will only be two factors:  `- 1` and the number itself.
• For a composite number there will be more than two factors.
• Division and multiplication are the primary operations used to determine the factors of a number.

How to Find Factors of a Number?

There are two possible methods to find factors of any number, such as:

• Multiplication
• Division

Finding Factors: Multiplication Method

Let’s look into an example to understand how to determine the factors of a number using the multiplication method.

Example: Find the factors of `48` using the multiplication method.

Solution: Let us find the factors of `48` by multiplication method using the following steps.

Step `1`: To find the factors of `48` using multiplication, begin by dividing `48` by natural numbers starting from `1` and continuing until `10`. 

Step `2`: As you divide and list all numbers up to `10`, note pairs of numbers that multiply to give `48`. For instance, starting with `1`, note `1 × 48 = 48` and `2 × 24 = 48`. Continue this process to form pairs such as `(1, 48), (2, 24),` and so on.

Step `3`: After checking numbers up to `10` and noting the pairs, organize the factors of `48` starting from `1` listing the factors downwards, then repeat the process going up to `48`. This provides a comprehensive list of all factors of `48`.

Thus, the factors of `48` are `1, 2, 3, 4, 6, 8, 12, 16, 24` and `48`.

Note: We do not continue beyond `10` because the pairs start repeating beyond that.

Finding Factors: Division Method

Let’s find the factors of a number using the division method.

Example: Find the factors of `24` using the division method.

Solution: Let us find the factors of `24` by division method using the following steps.

Step `1`: To find the factors of `24` using the division method, begin by checking which numbers divide `24` completely. Start with `1` and record all the numbers that divide `24` without leaving a remainder.

Step `2`: The numbers that divide `24` completely are its factors. Write each of these numbers along with its pair. This can be done by performing division, such as `24 ÷ 1 = 24` and `24 ÷ 2 = 12`. Each divisor and quotient pair represents a factor of `24`. For example, `(1, 24)` is the first pair, `(2, 12)` is the second pair, and so on. By listing the factors in this manner, both factors in each pair are identified.

Step `3`: Once all possible divisors are checked and listed, organize the factors of `24`. Start from `1` at the top, list the factors downwards, then repeat the process going up to `24`. This provides a comprehensive list of all the factors of `24`.

Thus, the factors of `24` can be listed as `1, 2, 3, 4, 5, 6, 8, 12` and `24`.

Finding the Number of Factors

We can determine the number of factors for a given number by following these steps.

Step `1`: Begin by finding the prime factorization of the given number, breaking it down into its prime factors.

Step `2`: Express the prime factorization in exponent form.

Step `3`: Increase each exponent by `1`.

Step `4`: Multiply all the resulting numbers together. This product yields the total number of factors for the given number.

Example: Determine the number of factors of \(180\).

Solution: 

Let's determine the number of factors of \(180\) by following the steps outlined below.

Step `1`: Prime factorization of \(180\) gives us \(180 = 2 \times 2 \times 3 \times 3 \times 5\).

Step `2`: Expressing the prime factorization in exponent form gives us \(180 = 2^2 \times 3^2 \times 5^1\).

Step `3`: Adding \(1\) to each exponent, we obtain \((2 + 1) = 3, (2 + 1) = 3,\) and \((1 + 1) = 2\).

Step `4`: Multiplying these results gives us \(3 \times 3 \times 2 = 18\). Therefore, \(180\) has \(18\) factors.

We can validate this number by listing the actual factors of \(180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90,\) and \(180\). Thus, we have verified that \(180\) has \(18\) factors.

Factor Pairs

A factor pair consists of two numbers that, when multiplied together, yield a specific product. A number can have multiple factor pairs. These pairs can be positive or negative but not fractional or decimal. Let us understand this with the help of an example.

Example: Find the factor pairs of `32`.

Solution: Let us find the factor pairs of `32` using the following steps.

Step `1`: Begin by listing all the numbers that divide evenly into `32`. These are `1, 2, 4, 8, 16,` and `32`.

Step `2`: Next, pair up these numbers so that when multiplied together, they equal `32`. For example, `1 × 32 = 32`, `2 × 16 = 32`, and `4 × 8 = 32`.

Step `3`: These pairs are the factor pairs of `32`, which can be expressed as `(1, 32), (2, 16),` and `(4, 8)`. 

This shows that no matter which pair of factors you choose, their product will always be `32`.The factors of `32` in pairs can be written as shown in the table given below:

Negative factor pairs of a number are possible because multiplying two negative numbers results in a positive product. Consider the negative factor pairs of `32`.

Solved Examples

Example `1`: What are factors of `81`?

Solution:

• To find the factors of `81` using the multiplication method, we start by dividing `81` by natural numbers starting from `1` and progressing up to `9`. 
• We observe pairs of numbers that, when multiplied, result in `81`. These pairs include `1 × 81 = 81, 3 × 27 = 8,` and `9 × 9 = 81`. 
• By listing these pairs in ascending and then descending order, we compile a complete list of all the factors of `81`. 
• Therefore, the positive factors of `81` are `1, 3, 9, 27,` and `81`.

Example `2`: Find the factor pairs of `19`.

Solution:

`19` is a prime number. The only two numbers that divide `19` completely are `1` and `19`.

So, the factors of `19` are `1` and `19`.

Therefore, the factor pair of `19` is `(1, 19)`.

Example `3`: List the factor pairs of `72`.

Solution:

To find the factor pairs of `72`, follow these steps:

Step `1`: Begin by listing all the factors of `72`, which are `1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36,` and `72`.

Step `2`: Next, pair these factors so that when multiplied together, they equal `72`. For instance, you can pair `1` with `72, 2` with `36, 3` with `24, 4` with `18,` and `6` with `12` and so on.

Step `3`: These pairings provide the factor pairs of `72`, which can be expressed as `(1, 72), (2, 36), (3, 24), (4, 18), (6, 12),` and `(8, 9)`.

Example `4`: What is the factor of `48` that is also an odd prime number.

Solution:

Let’s first list the factors of `48`.

• To find the factors of `48` using the multiplication method, we start by dividing `48` by natural numbers starting from `1` and progressing up to `9`. 
• We observe pairs of numbers that, when multiplied, result in `48`. These pairs include `1 × 48 = 48, 2 × 24 = 48, 3 × 16 = 48, 4 × 12 = 48,` and `6 × 8 = 48`. 
• By listing these pairs in ascending and then descending order, we compile a complete list of all the factors of `48`. 
• Therefore, the positive factors of `48` are `1, 2, 3, 4, 6, 8, 12, 16, 24,` and `48`.

In the given list, the only odd numbers are `1` and `3` of which `3` is a prime number. Hence the factor of `48` which is also an odd prime number is `3`.

Practice Problems

Q`1`. Find the factor pairs of `37`.

1.  `(1, 37)`  
2.  `(2, 18)`  
3.  `(1, 31)`  
4.  `(4, 9)`   

Answer: a

Q`2`. Find the factors of `50`.

1.  `2, 5, 10, 25, 50`
2. `1, 2, 5, 10, 25, 50`
3.  `1, 2, 5, 10, 25`
4.  `1, 2, 5, 10`

Answer: b

Q`3`. Which of the following is one of the factor pairs of `60`.

1.  `(1, 50)`  
2.  `(2, 30)`  
3.  `(30, 30)`  
4.  `(4, 20)`   

Answer: b

Q`4`. List the factors of `120`.

1.  `1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120`
2.  `1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60`
3.  `1, 2, 3, 4, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120`
4.  `1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 50, 60, 120`

Answer: a

Q`5`. Find the factors of `100`.

1.  `2, 4, 5, 10, 20, 25, 50, 100`
2.  `1, 2, 4, 5, 10, 20, 25, 50, 100`
3.  `1, 2, 4, 5, 10, 20, 25, 50`
4.  `1, 2, 4, 5, 10, 20, 25`

Answer: b

Frequently Asked Questions

Q`1`. What is a factor of a number? 

Answer: Factors are numbers that divide a given number evenly without leaving a remainder.

Q`2`. How do I find factors? 

Answer: Factors are found by dividing the number by other natural numbers and noting the pairs that result in whole numbers.

Q`3`. What is the significance of factors? 

Answer: Factors help in understanding the divisibility and mathematical properties of a number.

Q`4`. Can a number have infinite factors? 

Answer: No, every number has a finite number of factors.

Q`5`. Are `1` and the number itself always factors? 

Answer: Yes, every number is divisible by `1` and itself, making them automatic factors.