Let us first understand the meaning of factors. The factors of a number \(x\) are the numbers that divide \(x\)
with zero as a remainder. It means that if \(y\) divides \(x\) and the remainder is \(0\) then we say that \(y\) is a
factor of \(x\).
The factors of \(\textbf{225}\) are the numbers that can be multiplied together to get the product \(225\). In other words, the factors of \(225\) are the numbers that divide \(225\) without leaving a remainder. The factors of \(225\) are \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\).
In this mathematics article, we will learn about the factors of \(225\), how to find the factors of \(225\) by different methods such as prime factorization method, factor tree method, etc, and where it can be used with some solved examples.
What are the Factors of 225?
The factors of \(\textbf{225}\) are the numbers that can be multiplied together to get the product \(225\). In
other words, the factors of \(225\) are the numbers that divide \(225\) without leaving a remainder. The factors of
\(225\) are \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\). These are the only positive integers that can be
multiplied together in different ways to get the product \(225\).
To check why \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\) are the factors of \(225\), we can perform a simple division. When we divide \(225\) by \(1\), we get \(225\) with no remainder. When we divide \(225\) by \(3\), we also get \(75\) with no remainder. Similarly, check for other factors of \(225\).
Therefore, \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\) are the only factors of \(225\).
Prime Factors of 225:
Prime numbers are all positive integers that can only be evenly divided by \(1\) and itself. Prime factors of \(225\) are all the prime numbers that when multiplied together equal \(225\). We know that \(225\) is not a prime number, but it can be expressed as the product of prime numbers.
The process of finding the prime factors of \(225\) is called prime factorization of \(225\). To get the prime factors of \(225\), divide \(225\) by the smallest prime number possible. Then take the result from that and divide that by the smallest prime number. Repeat this process until you end up with \(1\), as shown in the figure.
So, the prime factorization of \(225\) is \(5 \times 5 \times 3 \times 3\). In other words, 225 as a product of prime factors is \(5 \times 5 \times 3 \times 3\). Therefore, the prime factors of \(225\) are \(3\), and \(5\).
Composite Factors of 225:
Composite numbers can be defined as numbers that have more than two factors. Numbers that are not prime are composite numbers because they are divisible by more than two numbers.
We know that the factors of \(225\) are \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\). Composite factors of \(225\) are \(9, 15, 25, 45, 75\), and \(225\). A number can be classified as prime or composite depending on their divisibility.
The number \(225\) is divisible by \(5\). So, we can say that \(225\) is a composite number and will surely have more than two factors. Similarly, we can check for other factors of \(225\). Therefore, the composite factors of \(225\) are \(9, 15, 25, 45, 75\), and \(225\).
Pair Factors of 225:
Pair factors of a number are the pairs of two numbers that when multiplied together give the original number. \(225\) can be expressed as a product of two numbers in all possible ways. In each product, both multiplicands are the factors of \(225\).
The table below shows the factor pairs of \(225\):
Factors | Pair Factors |
\(1 \times 225 = 225\) | \((1, 225)\) |
\(3 \times 75 = 225\) | \((3, 75)\) |
\(5 \times 45 = 225\) | \((5, 45)\) |
\(9 \times 25 = 225\) | \((9, 25)\) |
\(15 \times 15 = 225\) | \((15, 15)\) |
Therefore, from the above table we see that - \((1, 225)\), \((3, 75)\), \((5, 45)\), \((9, 25)\), and \((15, 15)\) are the only pair factors of \(225\).
Similarly, we can find the negative factor pairs of \(225\) as follows:
Factors | Negative Factor Pairs |
\(-1 \times -225 = 225\) | \((-1, -225)\) |
\(-3 \times -75 = 225\) | \((-3, -75)\) |
\(-5 \times -45 = 225\) | \((-5, -45)\) |
\(-9 \times -25 = 225\) | \((-9, -25)\) |
\(-15 \times -15 = 225\) | \((-15, -15)\) |
Therefore, from the above table we see that negative factor pairs of \(225\) are \((-1, -225)\), \((-3, -75)\), \((-5, -45)\), \((-9, -25)\), and \((-15, -15)\).
Common Factors of 225:
Common factors of two or more numbers are the numbers that divide both the numbers leaving zero as the remainder. The common factors of \(225\) are the factors that \(225\) shares with another number. Let us understand this with the help of an example.
Example: Find the common factors of \(65\) and \(225\).
First write the factors of \(65\) and the factors of \(225\).
Factors of \(65\) = \(1, 5, 13\), and \(65\).
Factors of \(225\) = \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\).
So, the common factors of both the numbers are \(1\) and \(5\).
Steps to find Factors of 225
Let us understand how to find the factors of \(225\) using the below steps:
Step 1: Start by dividing \(225\) by the smallest positive integer greater than \(1\).
Step 2: Check each integer to see if it divides evenly into \(225\), this means that it is a factor of \(225\).
Step 3: The only positive integers that divide evenly into \(225\) are \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\).
Therefore, these are the factors of \(225\).
How to Find the Factors of 225?
Let's learn to find all the factors of \(225\). We can find the factors of \(225\) by using below methods:
- Prime factorization of \(225\).
- Factor tree method to find factors of \(225\).
- Division method to find factors of \(225\).
Prime Factorisation of 225:
Prime factorization is the process of expressing a composite number as a product of its prime factors. A composite number is any positive integer greater than \(1\) that is not a prime number.
To find the prime factorization of \(225\), we need to find the prime factors of \(225\) and express it as a product of those prime factors. Follow the steps:
Step 1: To find the prime factors of \(225\), we start by dividing it by the smallest prime number, which is \(2\). However, \(2\) does not divide evenly into \(225\).
Step 2: Next, we try dividing \(225\) by the smallest prime number, which is \(5\). Since \(5\) divides evenly into \(225\), we can write: \(\frac{225}{5}=45\).
Step 3: Again, divide \(45\) by the smallest prime factor, i.e., \(5\) as \(\frac{45}{5}=9\).
Step 4: Again, divide \(9\) by the smallest prime factor, i.e., \(3\) as \(\frac{9}{3}=3\).
Step 5: Again, divide \(3\) by the smallest prime factor, i.e., \(3\) as \(\frac{3}{3}=1\).
Step 6: Now, we cannot divide \(1\) by any prime factor.
Therefore, the prime factorization of \(225 = 5 \times 5 \times 3 \times 3. Prime factorization of 225 using exponents can be written as 5^{2} \times 3^{2}\).
Factor Tree Method to Find Factors of 225:
The factor tree method can be a useful way to visually see the prime factors of a number and to find all of the factors. The method can also be extended to larger numbers by continuing to factor each factor until only prime numbers are left.
Here are the steps to use the factor tree method to find the prime factors of \(225\):
Step 1: To find the factors of \(225\) using the factor tree method, we can start by writing \(225\) at the top of the tree.
Step 2: First, select the factor pair with the smallest prime number. Here, we can take it as \(3\) and pair it up with \(75\), as \(3\) multiplied by \(75\) gives \(225\).
Step 3: \(3\) is a prime number, so it will remain unchanged and we can further split \(75\) into smaller prime factors. \(75\) can be further expressed as \(3\) multiplied by \(25\).
Step 4: \(3\) is a prime number, so it will remain unchanged and we can further split \(25\) into smaller prime factors. \(25\) can be further expressed as \(5\) multiplied by \(5\).
Step 5: As both the numbers in the last step are prime, i.e. \5\) and \(5\), we will terminate the splitting process.
Step 7: Bringing the prime factors all together, we get \(3, 3, 5\), and \(5\). Also, the product of \(3, 3, 5\), and \(5\) is equivalent to \(225\).
Therefore, by the factor tree method, the prime factors of \(225\) are \(3\), and \(5\).
Division Method to Find Factors of 225:
The division method is a systematic way of finding all the factors of a number. To use the division method to find the factors of \(225\), follow these steps:
Step 1: When we divide \(225\) by \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\), the remainder will be \(0\).
Step 2: At the same time, when we divide \(225\) by numbers like \(2, 4\) or \(6\), it leaves a remainder.
Step 3: Try to divide \(225\) by the above numbers and see the results.
Therefore, the factors of \(225\) are \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\) by division method.
Factors of 150 and 225
- The factors of \(150\) are the numbers that can be multiplied together to get the product \(150\). In other words, the factors of \(150\) are the numbers that divide \(150\) without leaving a remainder. The factors of \(150\) are \(1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75\), and \(150\).
- The factors of \(225\) are the numbers that can be multiplied together to get the product \(225\). In other words, the factors of \(225\) are the numbers that divide \(225\) without leaving a remainder. The factors of \(225\) are \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\).
- The common factors of \(150\) and \(225\) are \(1, 3, 5, 15, 25\), and \(75\).
Factors of 225 Summary
- The factors of \(225\) are the numbers that can be multiplied together to get the product \(225\).
- The factors of \(225\) are \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\).
- The prime factors of \(225\) are \(3\), and \(5\).
- The negative factors of \(225\) are \(-1, -3, -5, -9, -15, -25, -45, -75\), and \(-225\).
- The positive factor pairs of \(225\) are \((1, 225)\), \((3, 75)\), \((5, 45)\), \((9, 25)\), and \((15, 15)\).
- The negative factor pairs of \(225\) are \((-1, -225)\), \((-3, -75)\), \((-5, -45)\), \((-9, -25)\), and \((-15, -15)\).
- The prime factorization of \(225\) is \(225 = 3^{2} \times 5^{2}\).
- Sum of factors of \(225\) is \(403\), i.e. \(1 + 3 + 5 + 9 + 15 + 25 + 45 + 75 + 225 = 403\).
Factors of 225 Solved Examples
- Find the common factors of 15 and 225.
Solution:
The factors of \(15\) are \(1, 3, 5\), and \(15\), while the factors of \(225\) are \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\).
So, the common factors of \(15\) and \(225\) are \(1, 3, 5\), and \(15\).
- What is the sum of all the factors of 225?
Solution:
The factors of \(225\) are \(1, 3, 5, 9, 15, 25, 45, 75\), and \(225\).
Sum \(= 1 + 3 + 5 + 9 + 15 + 25 + 45 + 75 + 225 = 403\)
Therefore, \(403\) is the required sum.
- If there are 225 mangoes to be distributed among 15 children. How many mangoes does each child get?
Solution:
Given,
Number of mangoes = \(225\)
Number of children = \(15\)
Each child will get = \(\frac{225}{15} = 15\) mangoes
Therefore, each children will get \(15\) mangoes each.
- Find the HCF of 135 and 225, and HCF of 867 and 225.
Solution:
(i). The HCF of \(135\) and \(225\) is the largest possible number that divides \(135\) and \(225\) exactly without any remainder.
First write the factors of \(135\) and \(225\):
Factors of \(135\): \(1, 3, 5, 9, 15, 27, 45, 135\).
Factors of \(225\): \(1, 3, 5, 9, 15, 25, 45, 75, 225\).
There are \(6\) common factors of \(135\) and \(225\), that are \(1, 3, 5, 9, 15\), and \(45\).
Therefore, the highest common factor (HCF) of \(135\) and \(225\) is \(45\).
(ii). The HCF of \(867\) and \(225\) is the largest possible number that divides \(867\) and \(225\) exactly without any remainder.
First write the factors of \(867\) and \(225\):
Factors of \(867\): \(1, 3, 17, 51, 289, 867\).
Factors of \(225\): \(1, 3, 5, 9, 15, 25, 45, 75, 225\).
There are \(2\) common factors of \(867\) and \(225\), that are \(1\) and \(3\).
Therefore, the highest common factor (HCF) of \(867\) and \(225\) is \(3\).
We hope that the above article is helpful for your understanding and exam preparations. Stay
tuned to the Testbook App for more updates on related topics from Mathematics, and various
such subjects. Also, reach out to the test series available to examine your knowledge
regarding several exams.