# The Prime Factorization of 42 Explained

The prime factorization of a number involves breaking it down into its prime factors. These are the numbers that can only be divided by 1 and themselves, so they are the building blocks of every other number. Prime factorization is an important concept in mathematics, as it helps us understand how numbers work and how they can be used in different applications.

In this article, we will explain the prime factorization of the number 42. This is a relatively small number, so it is a good place to start if you are new to prime factorization. We will begin by defining what prime factorization is, and then we will show you how to find the prime factors of 42.

Prime factorization is the process of breaking down a number into its prime factors. A prime factor is a prime number that divides the original number without leaving a remainder. For example, the prime factors of 6 are 2 and 3, because 2 x 3 = 6 and these are the only two prime numbers that can be multiplied together to get 6.

To find the prime factors of 42, we need to start by dividing it by the smallest possible prime number, which is 2. We write the division as follows:

42 ÷ 2 = 21

This means that 2 is a factor of 42, because it divides into it exactly 21 times. Now we need to divide 21 by another prime number, and the next smallest prime number is 3. We write the division as follows:

21 ÷ 3 = 7

This means that 3 is also a factor of 42, because it divides into it exactly 7 times. Now we have a prime factorization for 42: 2 x 3 x 7. These are the only prime factors of 42, because they are the smallest prime numbers that can be multiplied together to get 42.

Another way to write the prime factorization of 42 is to use exponents. This involves writing each prime factor once and then adding an exponent to show how many times it appears in the factorization. For example, we can write the prime factorization of 42 as:

2^1 x 3^1 x 7^1

This means that 2, 3, and 7 each appear once in the factorization, so we write a 1 as the exponent for each prime number.

The prime factorization of 42 has several important applications in mathematics. For example, it can be used to find the greatest common divisor (GCD) and least common multiple (LCM) of two or more numbers. To find the GCD of two numbers, we need to find their prime factorizations and then multiply together the common prime factors with the smallest exponents. For example, if we want to find the GCD of 42 and 56, we first write their prime factorizations:

42 = 2^1 x 3^1 x 7^1
56 = 2^3 x 7^1

The common prime factors are 2 and 7, so we take the product of 2^1 x 7^1, which is 14. Therefore, the GCD of 42 and 56 is 14.

To find the LCM of two numbers, we need to find their prime factorizations and then multiply together all the prime factors with the highest exponent. For example, if we want to find the LCM of 42 and 56, we first write their prime factorizations:

42 = 2^1 x 3^1 x 7^1
56 = 2^3 x 7^1

We take the product of 2^3 x 3^1 x 7^1, which is 2 x 2 x 2 x 3 x 7 = 168. Therefore, the LCM of 42 and 56 is 168.

In conclusion, the prime factorization of 42 is 2 x 3 x 7. This means that 2, 3, and 7 are the only prime factors of 42, and they can be multiplied together to get 42. Prime factorization is an important concept in mathematics, as it helps us understand how numbers work and how they can be used in different applications. By finding the prime factors of a number, we can also find its GCD and LCM, which are useful in many areas of mathematics and science.