Factors of 225: Prime Factorization, Methods, Tree, and Examples

The factors of 225 comprise the set of natural numbers which give 225 as the whole number product when these numbers are subjected to multiplication with each other. The numbers pair up together, giving 225 as the product to form a factor pair.

Factors of two twenty five

Figure 1 – All possible Factors of 225

In another definition, the factors of 225 can also be categorized as those numbers which produce a zero remainder and a whole number quotient when 225 acts as the dividend for such numbers.

The number 225 is an odd composite number. Since the number is a composite number, this indicates that 225 will consist of more than two factors. The factors consist of both positive and negative values.

These factors of 225 can be evaluated through multiple processes, but the most common ones are the division method and the prime factorization. The prime factors that are determined can then also be arranged in the form of a factor tree.

In this article, we will be taking a very detailed overview of the factors of 225. We will go through all the necessary methods utilized for determining these factors. And lastly, we will take a look at some solved examples of factors of 225.

What Are the Factors of 225?

The factors of 225 are 1, 3, 5, 9, 15, 25, 45, 75, and 225. In total, the number 225 consists of 9 factors, which form 5 factor pairs.

The number 15 exists twice in the factor list because the number 225 is a perfect square of the number 15. All these numbers yield a zero remainder. 

How To Calculate the Factors of 225?

You can calculate the factors of the number 225 through two methods – the division method and the prime factorization method. The division method determines the prime and the composite factors but the prime factorization method only determines the prime factors.

Division

Let’s first take a look at the basic method, which is the division method.

The division method is simple and straightforward, based on the division operation. The division method states that if a divisor successfully yields a whole number quotient and a zero remainder, then that divisor is qualified to be a factor.

Keeping this foundation of the division method in mind, let’s move on to the application of the division method on the number 225.

It is obvious by analyzing the number 225 that it is an odd number that rules out two from being a factor of 225. Let’s look at the various divisions of numbers that act as the factors of 225.

\[ \frac{225}{1} = 225 \]

\[ \frac{225}{3} = 75 \]

\[ \frac{225}{5} = 45 \]

\[ \frac{225}{9} = 25 \]

\[ \frac{225}{15} = 15 \]

\[ \frac{225}{25} = 9 \]

\[ \frac{225}{45} = 5 \]

\[ \frac{225}{75} = 2 \]

\[ \frac{225}{225} = 1\]

Therefore, the definitive list of factors of 225 is:

Factors of 225: 1, 3, 5, 9, 15, 25, 45, 75, and 225

These factors are also capable of existing in the form of negative numbers. The condition for negative factors is that they must be paired up with a negative sign. Keeping this condition in mind, the negative factors of 225 are given below:

Negative Factors of 225: -1, -3, -5, -9, -15, -25, -45, -75, and -225

Factors of 225 by Prime Factorization

Prime Factorization is one of the techniques which is used to determine the factors of a number. To be specific, the prime factorization method is used to figure out the prime factors of a number.

The prime factorization technique is based on the mathematical operation of division. It begins with the number as the dividend and prime numbers as divisors. The place of the dividend keeps on getting replaced with the whole number quotient in every step.

This division via prime numbers continues until only one remains in the end. This concludes the prime factorization technique.

The prime factorization of the number 225 is shown below:

225 $\div$ 5 = 45

45 $\div$ 5 = 9 

9 $\div$ 3 = 3

3 $\div$ 3 = 1

The prime factorization of 225 can be written as:

Prime Factorization of 225 = 5 x 5 x 3 x 3

OR

Prime Factorization of 225 = $5^{2}$ x $3^{2}$

The prime factorization of 225 is depicted in figure 1 given below:

Prime factorization of two hundred and twenty five

Figure 2 – Prime Factorization of 225

Hence, the prime factors obtained are:

Prime Factors of 225 = 3 and 5 

Factor Tree of 225

A Factor Tree is a tree-like diagram representing any number’s prime factors. The factor tree consists of a root number and two branches that keep on expanding at every step. 

The factor begins with the base number, which is the number itself, in this case, 225. This number splits into two branches. The division process in the factor tree is identical to the one in prime factorization; the only difference between the two is that prime factorization concludes on one, whereas the factor tree concludes on prime numbers.

The factor tree for the number 225 is given below in figure 2:

Factor tree of two hundred and twenty five

Figure 3 – Factor Tree of 225

Factors of 225 in Pairs

The nine factors of the number 225 can also exist in the form of pairs. The factor pairs are the numbers that generate the original number as the product when these two factors are multiplied by each other.

The factor pair consists of 2 numbers within a pair. These two numbers undergo the process of multiplication with each other.

The factor pairs of the number 225 are given below:

Factors of 225: 1, 3, 5, 9, 15, 25, 45, 75, and 225

Formulating the factor pairs:

1 x 225 = 225

3 x 75 = 225

5 x 45 = 225

9 x 25 = 225

15 x 15 = 225

So, the factor pairs of 225 are:

Factor Pairs of 225: (1, 225), (3, 75), (5, 45), (9, 25), and (15, 15)

As the factor of 225 can be negative, the factor pairs can also exist in the form of negative numbers. The condition for these negative factor pairs is that both the numbers must have a negative sign, so when they are multiplied together, they produce a positive product.

The negative factor pairs of the number 225 are given below:

-1 x -225 = 225

-3 x -75 = 225

-5 x -45 = 225

-9 x -25 = 225

-15 x -15 = 225

Negative Factor Pairs of 225: (-1, -225), (-3, -75), (-5, -45), (-9, -25), and (-15, -15)

Factors of 225 Solved Examples

To learn more about the factors of 225, let’s take a look at some solved examples involving these factors of 225. 

Example 1

Separate the odd and even factors of the number 225. Calculate their averages separately.

Solution

To begin with, the solution of this example, let’s first list these factors down:

Factors of 225 = 1, 3, 5, 9, 15, 25, 45, 75, and 225

Now, let’s separate the even factors. Even factors are the numbers that are a multiple of 2. 

Even Factors of 225 = There are no even factors of 225

So, the average of even factors of 225 is:

Average of Even Factors of 225 = 0

Now, let’s note down the odd factors of 225:

Odd Factors of 225 = 1, 3, 5, 9, 15, 25, 45, 75, and 225

The formula for average is:

\[ \text{Average of odd factors of 225} = \frac{\text{Sum of odd factors}}{\text{Total number of odd factors}} \]

\[ \text{Average of odd factors of 225} = \frac{1 + 3 + 5 + 9 + 15 + 25 + 45 + 75 + 225}{9} \]

\[ \text{Average of odd factors of 225} = \frac{403}{9} \]

Average of odd factors of 225 = 44.778

So the average of odd factors of 225 is 44.778.

Example 2

Calculate the sum of the factors of 225 and the product of factors of 225. 

Solution

Let’s start the solution by first listing down the factors of 225:

Factors of 225: 1, 3, 5, 9, 15, 25, 45, 75, and 225

Now, let’s calculate the sum of these factors:

Sum of factors of 225 = 1 + 3 + 5 + 9 + 15 + 25 + 45 + 75 + 225

Sum of factors of 225 = 403

So the sum of the factors of 225 is 403.

Now, let’s calculate the product of the factors of 225.

Product of factors of 225 = 1 x 3 x 5 x 9 x 15 x 25 x 45 x 75 x 225

Product of factors of 225 = 38, 443, 359, 375

Hence, we have determined the sum and the product of the factors of 225.

All images/mathematical drawings are created with GeoGebra.

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