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Factors of -40: Prime Factorization, Method, Tree, and Examples

Factors of -40 include the numbers that evenly divide -40 having zero remainders. If the remainder is a non-zero number, it will not be considered in the list of factors. 

-40 has both positive and negative factors. If the factor pair has both numbers positive, the product will be a positive number, and if both numbers are negative again, the product will be positive. The product will be negative only if the factor pair has one positive number and another one should be a negative number. This is also known as multiplication law

In this article, we will learn what are the factors of -40, and different methods to find them. There are also some solved examples for better understanding. 

What Are the Factors of -40?

The factors of  -40 are 1, -1, 2, -2, 4, -4, 5, -5, 8, -8, 10, -10, 20, -20,  40, and -40. These integers are included in the list of factors of -40 as they divide -40 by leaving the remainder zero.

-40 has sixteen factors in total. By multiplying these integers in pairs such that the product is equal to -40, then these numbers are said to be the factors of -40.

How To Calculate the Factors of -40?

You can calculate the factors of -40 by using the rules of divisibility that demand the remainder to be zero for a number to be in the list of factors of the given number.

There are two methods to calculate the factors:

  1. Division Method.
  2. Multiplication Method.

In the multiplication method, we will follow the law of multiplication. Factor pairs have both positive and negative numbers as their entry, which results in a negative number as a product. In the division method rules of division will be followed.

-40 is not a prime number. It will have more than two factors. To find factors of -40, simply start dividing it by different numbers and check for both positive and negative numbers. If the remainder is zero consider it as a factor of -40. 

Number 1 is a factor of every whole number. As a result 1 and -1, both are factors of -40. 

-40 is an even number, so it can be divided by 2 and -2

\[\frac {-40}{2}= -20\]

\[\frac {-40}{-2}= 20\]

2 is a positive factor and -2 is a negative factor of -40.

Dividing -40 by 3 results in a non-zero remainder:

\[\frac {-40}{3}= -13.3\]

The remainder is -1, which is a non-zero number, so 3 can not be a factor of -40.

Dividing -40 by 4 and -4 gives:

\[\frac {-40}{4}= -10\]

\[\frac {-40}{-4}= 10\]

The remainder is zero, so 4 and -4 are also the factors of -40.

As we know -40 is a multiple of 5, 8, 10, and 20 therefore, it is divisible by 5, -5, 8, -8, 10, -10, 20, and-20 which means the remainder will be zero. 

\[\frac {-40}{5}= -8\]

\[\frac {-40}{-5}= 8\]

\[\frac {-40}{8}= -5\]

\[\frac {-40}{-8}= 5\]

\[\frac {-40}{10}= -4\]

\[\frac {-40}{-10}= 4\]

\[\frac {-40}{20}= -2\]

\[\frac {-40}{-20}= 2\]

Hence, 5, -5, 8, -8, 10, -10, 20, and -20 are also the factors of -40.

The last factors will be the numbers 40 and -40 because every number divides itself fully.

\[\frac {-40}{40}= -1\]

\[\frac {-40}{-40}= 1\]

By the above calculations, we conclude that the factors of -40 are given as:

Factors of -40 = 1, -1, 2, -2, 4, -4, 5, -5, 8, -8, 10, -10, 20, -20,  40, -40 

Factors of -40 by Prime Factorization

Prime factorization means writing a number as a product of its prime factors. Factors that are prime in number are called prime factors.

Prime factorization can be done by dividing -40 by the smallest prime factor other than one, which will be 2. Again, divide the quotient by the smallest prime factor, if not divisible by 2 go for the next prime factor. Keep on dividing until the quotient becomes 1.

Prime factorization of -40 is shown below in figure 1:

Figure 1

The prime factorization of -40 is given as:

Separate the negative sign

\[ 2 \times 2 \times 2 \times 5 = 40 \]

Now multiply by the negative sign that we separated earlier.

\[ -1 \times 40 = -40 \]

Factor Tree of -40

The factor tree is a special diagram that expresses the prime factorization of a number. It consists of the factored number at the top; further, it splits into branches. Every branch contains factors. A factor tree is a pictorial representation.

The factor tree of -40 is shown below as:

Figure 2

We are dividing -40 into its factors. First of all, split -40 into 2 and -20, where 2 is the prime number, so it cannot be factored further. -20 has been further factorized into 2 and -10. Again, splitting -10 gives 2 and -5.

Factors of -40 in Pairs

Writing factors of a number in pairs such that their product is equal to the number itself. Such pairs are known as factor pairs.

Factor pairs of -40 are as follows:

\[ -1 \times 40= -40 \]

\[ 1 \times -40= -40 \]

\[ -2 \times 20= -40 \]

\[ 2 \times -20= -40 \]

\[ -4 \times 10= -40 \]

\[ 4 \times -10= -40 \]

\[ -5 \times 8= -40 \]

\[ 5 \times -8= -40 \]

When a negative sign is multiplied by a negative sign, their product is always positive. 

By looking at the above multiplication we will write the factor pairs for -40 as:

\[ (-1, 40) \]

\[ (1, -40) \]

\[ (-2, 20) \]

\[ (2, -20) \]

\[ (-4, 10) \]

\[ (4, -10) \]

\[ (-5, 8) \]

\[ (5, -8) \]

Factors of -40 Solved Examples

Let’s solve some examples of factors of -40 for better understanding. 

Example 1

Anna has 8 as one of the factors of -40. Help her get the other factor of the pair.

Solution

Factor pair of -40: \[ Factor 1 \times Factor 2= -40 \]   

Factor 1: 8

By putting the value of Factor 1 in the above expression.

\[ 8 \times Factor 2= -40 \]

By rearranging the equation

\[\frac {-40}{8}= -5\]

Factor 2: -5

-5 will be the second factor of the pair.

(8, -5) is the factor pair of -40.

Example 2

Find the common factors of 500 and -40.

Solution

Factors of 500 are:

Factors of 500 = 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500 

Factors of -40 are:

Factors of -40 = 1, -1, 2, -2, 4, -4, 5, -5, 8, -8, 10, -10, 20, -20,  40, -40 

Common factors of 500 and -40 are 1, 2, 4, 5, 10, and 20.

Images/mathematical drawings are created with GeoGebra.

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