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# Factors of -48: Prime Factorization, Methods, Tree, and Examples

The **factors of -48** are the numbers that evenly divide -48 with zero remainders. -48 is a negative integer but its factors can be positive as well as negative. It cannot have the positive factor pairs as their product will result in a positive 48. But the factor pairs of -48 can contain 1 positive number and 1 negative number.

As the** laws of multiplication** conclude that when a negative number is multiplied by a positive number it results in a negative number. Therefore, one of the factors in the factor pair must be positive, and the other must be negative.

Let’s find out what are the factors of -48 and how to calculate them using various factorization techniques such as **prime factorization** and **division method**.

## What Are the Factors of -48?

**Factors of -48 are 1, -1, 2, -2, 3 -3, 4 -4, 6 -6, 8 -8, 12 -12, 16 -16, 24 -24, 48, and -48. All of these are regarded as the factors of -48 as their product is equivalent to -48.**

The **factors of -48** can be found using the division and multiplication method. In the division method rules of divisibility are used whereas in the multiplication method factor pairs are found which results in -48 as a product.

## How To Calculate the Factors of -48?

You can calculate the **factors of -48** by first determining whether it is a composite number or a prime number. Since -48 is a **composite number** therefore it has more than two factors and can undergo exact division to give a conclusive quotient. The divisor in the division is called the factor of -48.

-48 is also a negative even number. So, it can be divided by 2 as well as -2.

\[ \dfrac{-48}{2} = -24 \]

\[ \dfrac{-48}{-2} = 24 \]

So, 2 and -2 both are the factors of -48 regarded as positive and negative factor.

3 is also divisible by -48 given as:

\[ \dfrac{-48}{3} = -16 \]

\[ \dfrac{-48}{-3} = 16 \]

So 3 and -3 are also the factors of -48.

4 can also be divided by -48 as it is also an even number and the **divisibility rule for 4** states that if the last 2-digits of the given number are divisible by 4 then the given number is also divisible by 4. Hence 48 is divisible by 4.

\[ \dfrac{-48}{4} = -12 \]

\[ \dfrac{-48}{-4} =12\]

Therefore, 4 and -4 are the factors of -48.

Coming forward -48 is also the **multiple of 6, 8, and 12** therefore it can be divided by 6, 8, and 12 as well leaving nothing behind as a remainder.

\[ \dfrac{-48}{6} = -8 \]

\[ \dfrac{-48}{-6} = 8 \]

\[ \dfrac{-48}{8} = -6 \]

\[ \dfrac{-48}{-8} = 6 \]

\[ \dfrac{-48}{12} = -4 \]

\[ \dfrac{-48}{-12} = 4 \]

Hence 6, -6, 8, -8, 12, and -12 are also the **factors of -48**.

In a similar manner it can be presumed that all the integers divisible by -48 are the factors.

The remaining factors of -48 are given as:

\[ \dfrac{-48}{16} = -3 \]

\[ \dfrac{-48}{-16} = 3 \]

\[ \dfrac{-48}{24} = 2 \]

\[ \dfrac{-48}{-24} = -2 \]

Lastly, every number has 1 and itself as its factors. Therefore 1, -1, 48, and -48 are also factors of -48.

\[ \dfrac{-48}{1} = -48 \]

\[ \dfrac{-48}{-1} = 48 \]

\[ \dfrac{-48}{48}= -1 \]

\[ \dfrac{-48}{-48} = 1 \]

The **factor list -48** is given as:

**Factor List: 1, -1, 2, -2, 3 -3, 4 -4, 6 -6, 8 -8, 12 -12, 16,-16, 24 -24, 48, -48 **

## Factors of -48 by Prime Factorization

**Prime factorization** is the method of calculating prime factors of the given number -48. It is called prime factorization because every factor found using this is a prime number.

The **prime factorization of -48** can be done by dividing it by the smallest prime factor which is 2. Upon dividing -48 by 2, -24 is obtained. It can be divided further by 2 to yield 12 as a quotient.

Again further **division** is possible. Therefore keep dividing it by 2 till 1 is obtained as the result.

The prime factorization of -48 is shown in figure 1 below.

**Prime Factorization of -48 = -2 x -2 x -2 x -2 x -3 **

**$(-2)^4$ x -3 = -48 **

## Factor Tree of -48

The **factor tree** method gives us a pictorial representation of the factorization of -48 in terms of its **prime factors**. It is a way to represent the process of prime factorization. **Factor tree** comprises prime factors branches and each of them splits into factors of the number.

The **branch** of the factor tree always stops at the prime number which means that further division is not possible. The factor tree of -48 is shown below in figure 2:

## Factors of -48 in Pairs

**Factors** are written in pairs, which when multiplied together result in the original number.

The factor pairs of -48 are as follows:

**1 x -48 = -48 **

**2 x -24 = -48 **

**3 x -16 = -48 **

**4 x -12 = -48 **

**6 x -8 = -48 **

Therefore the **factor pairs** of -48 are the following:

**(1, -48)**

**(2, -24)**

**(3, -16)**

**(4, -12)**

**(6, -8)**

The **factor pairs** can also be written as:

**-1 x 48 = -48 **

**-2 x 24 = -48 **

**-3 x 16 = -48 **

**-4 x 12 = -48 **

**-6 x 8 = -48 **

Therefore the factor pairs are:

**(-1, 48)**

**(-2, 24)**

** (-3, 16)**

**(-4, 12)**

** (-6, 8)**

## Factors of -48 Solved Examples

Let’s solve some examples involving the factors of -48.

### Example 1

Find the range of the factors of -48.

### Solution

The factors of -48 are given as -48 -1, -2, -3, -4, -6, -8, -12, -16, -24, -48, 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

For finding the range write all these numbers in ascending order.

So the given list becomes:

**Ascending Order = -48, -24, -16, -8, -6, -4,****-3, -2, -1, 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 **

The formula for the range is given as:

**Range =largest value – smallest value **

**Range =48 – (-48)=96 **

### Example 2

Find the median of the positive factors of -48.

### Solution

As there are 10 positive factors of -48, the median can be found as the sum of all factors divided by the total number of factors.

\[Median= \dfrac{1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 + 48}{10} = 12.4 \]

*Images/mathematical drawings are created with GeoGebra.*