Understanding the Prime Factorization of 75

Prime factorization is a mathematical technique that is often used to break down a number into its prime factors. It is a fundamental concept in mathematics and has many practical applications, including cryptography, computer algorithms, and databases. In this article, we will explore the prime factorization of the number 75 and explain the process step by step.

Prime factorization is the process of finding the prime factors of a number. A prime number is a number that is only divisible by 1 and itself. For example, 2 is a prime number because it is only divisible by 1 and 2. However, 4 is not a prime number because it is divisible by 1, 2, and 4 (and other numbers).

To find the prime factors of 75, we first need to identify any prime numbers that can divide into 75. The first few prime numbers are 2, 3, 5, 7, 11, and so on. We can begin by trying to divide 75 by 2.

75 / 2 = 37.5

Since 37.5 is not a whole number, we know that 2 is not a factor of 75. We can move on to the next prime number, which is 3.

75 / 3 = 25

We have found a factor! 3 divides into 75 evenly, so we can write:

75 = 3 x 25

Now we need to factor 25 further. We can try dividing by 2 again, but we already know that 2 is not a factor of 75. We can try dividing by 3 instead.

25 / 3 = 8.33333

Again, we get a decimal, which means that 3 is not a factor of 25. We can try 5 next.

25 / 5 = 5

We have found another factor! 5 divides into 25 evenly, so we can write:

75 = 3 x 5 x 5

We have factored 75 into its prime factors: 3, 5, and 5.

The prime factorization of 75 is:

75 = 3 x 5 x 5

It is important to note that the order of the factors does not matter in the prime factorization. We could also write:

75 = 5 x 5 x 3

or

75 = 5 x 3 x 5

and they would still be correct prime factorizations of 75.

Prime factorization can also be done using a factor tree. A factor tree is a diagram that shows the prime factors of a number. To create a factor tree for 75, we start by dividing it by any prime factor. We can choose 3 or 5.

Let’s choose 3:

75

/ \

3 25

/ \

5 5

We have now factored 75 into its prime factors using a factor tree.

In conclusion, prime factorization is a useful mathematical technique that can be used to break down a number into its prime factors. To find the prime factors of a number, we need to test for divisibility by prime numbers until we can no longer divide evenly. The prime factors can then be multiplied together to get the prime factorization of the original number. Factor trees can also be used to visualize the prime factors of a number. By following these steps, we have successfully found the prime factorization of 75, which is 3 x 5 x 5.