What is the Prime Factorization of 126?

Learn how to find the prime factorization of 126 with this comprehensive guide. Discover the fundamental concept of prime factors and how to apply them in mathematics.

Introduction

Prime factorization is a fundamental concept in mathematics that breaks down a composite number into its prime factors. It’s an essential tool in solving various mathematical problems, including finding the greatest common factor, simplifying fractions, and solving equations. In this article, we’ll be discussing the prime factorization of 126.

Definition of Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors. A composite number is a positive integer greater than one that has at least one positive divisor other than one or itself. Prime factors are prime numbers that divide a composite number without leaving a remainder. In other words, prime factors are the building blocks of a composite number.

For instance, the prime factorization of 24 is 2 x 2 x 2 x 3. This means that 24 is the product of 2 raised to the power of 3 and 3 raised to the power of 1. By writing a number as a product of its prime factors, it becomes easier to work with and manipulate mathematically.

Now that we understand what prime factorization is, let’s dive into how to find the prime factorization of 126.

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How to Find Prime Factorization of 126

To find the prime factorization of 126, we can use the factor tree method. Here’s a step-by-step process to find the prime factorization of 126:

1. Start by dividing the number by the smallest prime number, which is 2.

2. Keep dividing by 2 until it’s no longer divisible by 2.

3. Then, divide the resulting number by the smallest prime factor, which is 3.

4. Keep dividing by 3 until it’s no longer divisible by 3.

5. Continue this process until the remaining number is itself a prime number.

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Using this method, we get the prime factorization of 126 as 2 x 3 x 3 x 7.

Another way to write the prime factorization of 126 is by using exponents. We can write 126 as 2 x 3² x 7, which means that 126 is the product of 2 raised to the power of 1, 3 raised to the power of 2, and 7 raised to the power of 1.

Factors of 126

To find the factors of 126, we need to find all the numbers that divide 126 without leaving a remainder. The factors of 126 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, and 126.

We can identify the prime factors of 126 by listing its factors and then selecting only the prime factors. The prime factors of 126 are 2, 3, and 7. This means that 126 is the product of 2 raised to the power of 1, 3 raised to the power of 2, and 7 raised to the power of 1.

Knowing the prime factorization of 126 and its factors can help us solve various mathematical problems, such as finding the greatest common factor and least common multiple of two or more numbers.

Now that we’ve discussed the prime factorization and factors of 126, let’s move on to writing the prime factorization of 126 using exponents.

Prime Factorization of 126

To find the prime factorization of 126, we need to first identify the prime factors of 126. We can do this by dividing 126 by the smallest prime number, which is 2.

126/2 = 63

We can see that 2 is not a factor of 63. Instead, 3 is the next smallest prime number.

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63/3 = 21

Again, 3 is not a factor of 21. We can divide 21 by 3 again.

21/3 = 7

Now, we have found the prime factors of 126, which are 2, 3, and 7. Therefore, the prime factorization of 126 is 2 x 3 x 3 x 7.

We can also write the prime factorization of 126 using exponents. To do this, we write each prime factor as a base with its exponent being the number of times it appears in the factorization.

2 x 3 x 3 x 7 can be written as 2^1 x 3^2 x 7^1.

Conclusion

In conclusion, prime factorization is a critical concept in mathematics that helps in solving various mathematical problems. In this article, we discussed the prime factorization of 126, which is 2 x 3 x 3 x 7 or 2^1 x 3^2 x 7^1. By breaking down composite numbers into their prime factors, we can simplify mathematical problems and make them more manageable.

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