Factors are the numbers that divide another number exactly, leaving no remainder. In simple terms, if a number divides another number completely, it is called a factor. For example, if you divide 12 by 3, the answer is 4 with no remainder. So, 3 is a factor of 12.

Now let’s look at the number 75. The numbers that divide 75 evenly are called its factors. These are:

1, 3, 5, 15, 25, and 75.

This means:

• 1 × 75 = 75
• 3 × 25 = 75
• 5 × 15 = 75

So, these pairs of numbers multiply to give 75, which confirms they are factors of 75. These numbers divide 75 completely, so no remainder is left. That’s what makes them factors!

In this mathematics article, we will learn about the factors of \(75\), how to find the factors of \(75\) 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 75?

Factors of 75 are the positive integers that divide 75 exactly, leaving no remainder. In simple terms, if a number divides 75 completely without anything left over, then that number is a factor of 75.

The factors of 75 are: 1, 3, 5, 15, 25, and 75.

Let’s check this:

• 75 ÷ 1 = 75
• 75 ÷ 3 = 25
• 75 ÷ 5 = 15
• 75 ÷ 15 = 5
• 75 ÷ 25 = 3
• 75 ÷ 75 = 1

All these give exact answers with no remainder. So, these are all the positive integers that divide 75 evenly. They are also the only numbers that, when multiplied in pairs, give the product 75 (like 3 × 25, 5 × 15). That’s why they are called the factors of 75.

Therefore, \(1, 3, 5, 15, 25\), and \(75\) are the only factors of \(75\).

Prime Factors of 75

Prime numbers are all positive integers that can only be divided by \(1\) and itself. Prime factors of \(75\) are all the prime numbers that when multiplied together equal \(75\). \(75\) is not a prime number, but it can be expressed as the product of prime numbers. 

The process of finding the prime factors of \(75\) is called prime factorization of \(75\). To get the prime factors of \(75\), divide \(75\) 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 \(1\).

So, the prime factorization of \(75\) is \(3 \times 5 \times 5\).

Therefore, the prime factors of \(75\) are \(3\), and \(5\).

Composite Factors of 75

Composite numbers can be defined as numbers that have more than the usual two factors; 1 and itself. Numbers that are not prime are composite numbers because they are divisible by more than two numbers.

We know that the factors of \(75\) are \(1, 3, 5, 15, 25\), and \(75\). Composite numbers in mathematics are the numbers that have proper factors i.e., factors other than 1 and the number itself.. Composite factors of \(75\) are \(15\) and \(25\). A number can be classified as prime or composite depending on their divisibility. 

The number \(25\) has an odd number at its unit's place, therefore it is divisible by \(5\). So, we can say that \(25\) is a composite number and will surely have more than two factors. Similarly, we check for other factors of \(75\). Therefore, the composite factors of \(75\) are \(15\) and \(25\).

Pair Factors of 75

Pair factors of a number are the pairs of two numbers that when multiplied together give the original number. \(75\) can be expressed as a product of two numbers in all possible ways. In each product, both multiplicands are the factors of \(75\).

Positive pair factors of 75 are two positive numbers that, when multiplied together, give the product 75. These pairs show how 75 can be broken into two factors. The positive pair factors of 75 are:
(1, 75), (3, 25), and (5, 15).
Each pair contains two positive integers that divide 75 exactly with no remainder.

The table below shows the factor pairs of \(75\):

Therefore, from the above table we see that - \((1, 75)\), \((3, 25)\), and \((5, 15)\) are the only pair factors of \(75\).

Negative pair factors of 75 are two negative numbers that multiply together to give the product +75. Since the product of two negative integers is positive, these pairs are also valid factors of 75. The negative pair factors of 75 are:
(-1, -75), (-3, -25), and (-5, -15).
Each pair consists of negative integers that divide 75 exactly with no remainder.

Similarly, we can find the negative factor pairs of \(75\) as follows:

Therefore, from the above table we see that negative factor pairs of \(75\) are \((-1, -75)\), \((-3, -25)\), and \((-5, -15)\).

Common Factors of 75

Common factors of two or more numbers are the numbers that divide both the numbers leaving zero as the remainder. Let us understand this with the help of an example.

Example: Find the common factors of \(70\) and \(75\). 

First write the factors of \(70\) and the factors of \(75\).

Factors of \(70\) = \(1, 2, 5, 7, 10, 14, 35\), and \(70\).

Factors of \(75\) = \(1, 3, 5, 15, 25\), and \(75\).

So, the common factors of both the numbers are \(1\) and \(5\).

Steps to find Factors of 75

Let us understand how to find the factors of \(75\) using the below steps:

Step 1: Start by dividing \(75\) by the smallest positive integer greater than \(1\).

Step 2: Check each integer to see if it divides evenly into \(75\), this means that it is a factor of \(75\).

Step 3: The only positive integers that divide evenly into \(75\) are \(1, 3, 5, 15, 25\), and \(75\).

Therefore, these are the factors of \(75\).

How to Find the Factors of 75?

We can find the factors of \(75\) by using below methods:

• Prime factorization of \(75\).
• Factor tree method to find factors of \(75\).
• Division method to find factors of \(75\).

Prime Factorisation of 75:

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 \(75\), we need to find the prime factors of \(75\) and express it as a product of those prime factors. Follow the steps:

Step 1: To find the prime factors of \(75\), we start by dividing it by the smallest prime number, which is \(3\). Here, \(3\) divides evenly into \(75\), we can write: \(\frac{75}{3}=25\).

Step 2: Next, we try dividing \(25\) by the smallest prime number, which is \(5\). Since \(5\) divides evenly into \(25\), we can write: \(\frac{25}{5}=5\).

Step 3: Again, divide \(5\) by the smallest prime factor, i.e., \(5\) as \(\frac{5}{5}=1\).

Step 4: Now, we cannot divide \(1\) by any prime factor. 

Therefore, the prime factorization of \(75 = 3 \times 5 \times 5\).

Factor Tree Method to Find Factors of 75:

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 factors of \(75\):

Step 1: To find the factors of \(75\) using the factor tree method, we can start by writing \(75\) 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 \(25\), as \(3\) multiplied by \(25\) gives \(75\).

Step 3: \(3\) 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 4: As both the numbers in the last step are prime, i.e. \(5\) and \(5\), we will terminate the splitting process. 

Step 5: Bringing the prime factors all together, we get \(3, 5\), and \(5\). Also, the product of \(3, 5\), and \(5\) is equivalent to \(75\). Therefore, the prime factors of \(75\) are \(3\), and \(5\).

Let us see how a factor tree of \(75\) looks like:

Division Method to Find Factors of 75:

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 \(75\), follow these steps:

Step 1: When we divide \(75\) by \(1, 3, 5, 15, 25\), and \(75\), the remainder will be \(0\).

Step 2: At the same time, when we divide \(75\) by numbers like \(2, 4\) or \(6\), etc it leaves a remainder.

Step 3: Try to divide \(75\) by the above numbers and see the results.

Therefore, the factors of \(75\) are \(1, 3, 5, 15, 25\), and \(75\) by division method.

Factors of 75 Summary

• The factors of \(75\) are the numbers that can be multiplied together to get the product \(75\).
• The factors of \(75\) are \(1, 3, 5, 15, 25\), and \(75\).
• The prime factors of \(75\) are \(3\), and \(5\).
• The negative factors of \(75\) are \(-1, -3, -5, -15, -25\), and \(-75\).
• The positive factor pairs of \(75\) are \((1, 75)\), \((3, 25)\), and \((5, 15)\).
• The negative factor pairs of \(75\) are \((-1, -75)\), \((-3, -25)\), and \((-5, -15)\).
• The prime factorization of \(75\) is \(75 = 3 \times 5 \times 5\).
• The sum of factors of \(75\) is \(124\), i.e. \(1 + 3 + 5 + 15 + 25 + 75 = 124\). 

Important Notes

• Prime factors are not the same as all factors. Prime factors are only the prime numbers that multiply to give the original number.
• If a number has an odd number of factors, it means the number is a perfect square.
• Every number has at least two factors: 1 and the number itself.
• There are no whole number factors between a number n and n/2 (except for the number itself).
• A number that has more than two factors is called a composite number.
• 1 is not a prime number, and 2 is the smallest prime number.

Factors of 75 Solved Examples

Example 1: Write \(75\) as a product of prime factors.

Solution:

To get the prime factors of \(75\), divide \(75\) by the smallest prime number possible. Then you take the result from that and divide that by the smallest prime number. Repeat this process until you end up with \(1\), as shown below:

So, \(75\) as a product of prime factors is \(75 = 3 \times 5 \times 5\).

Example 2:  Find the common factors of \(75\) and \(25\).

Solution:

First write the factors of \(75\) and the factors of \(25\).

Factors of \(75\) = \(1, 3, 5, 15, 25\), and \(75\).

Factors of \(25\) = \(1, 5\) and \(25\).

So, the common factors of both the numbers are \(1, 5\) and \(25\).

Example 3:  Write 60 as a product of prime factors.

Solution:
We start dividing 60 by the smallest prime number:

• 60 ÷ 2 = 30
• 30 ÷ 2 = 15
• 15 ÷ 3 = 5
• 5 ÷ 5 = 1

So, the prime factorization of 60 is: 60 = 2 × 2 × 3 × 5

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