what is the prime factorization of 200

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30-03-2025

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The prime factorization of a number is finding the prime numbers that multiply together to give you that original number. It's a fundamental concept in number theory, and learning how to find it is a valuable skill. Let's break down how to find the prime factorization of 200.

Understanding Prime Numbers

Before we begin, it's crucial to understand what a prime number is. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. Examples include 2, 3, 5, 7, 11, and so on. Numbers that aren't prime are called composite numbers.

Finding the Prime Factorization of 200

There are several ways to approach this. We'll use the factor tree method, a visual approach that's easy to follow.

Step 1: Start with the original number.

Write down the number 200.

Step 2: Find the smallest prime factor.

The smallest prime number is 2. Since 200 is an even number, it's divisible by 2.

200 ÷ 2 = 100

Now, we have 2 and 100. We'll continue breaking down 100.

Step 3: Continue factoring.

100 is also an even number, divisible by 2:

100 ÷ 2 = 50

We now have 2, 2, and 50. Let's break down 50.

50 is also divisible by 2:

50 ÷ 2 = 25

We now have 2, 2, 2, and 25. Notice that 25 is no longer divisible by 2.

Step 4: Find the next prime factor.

The next smallest prime number is 3. 25 is not divisible by 3. The next prime number is 5. And 25 is divisible by 5:

25 ÷ 5 = 5

We now have 2, 2, 2, 5, and 5. Both of these are prime numbers.

Step 5: Write the prime factorization.

We've reached the end! Our prime factorization of 200 is: 2 x 2 x 2 x 5 x 5.

This can also be written as: 2³ x 5²

Therefore, the prime factorization of 200 is 2³ x 5².

Alternative Methods

While the factor tree is visually appealing, you could also use division to achieve the same result:

1. Divide 200 by 2 repeatedly until you get an odd number: 200 ÷ 2 = 100; 100 ÷ 2 = 50; 50 ÷ 2 = 25
2. Then, divide the result (25) by its prime factors: 25 ÷ 5 = 5; 5 ÷ 5 = 1

This leads to the same prime factorization: 2 x 2 x 2 x 5 x 5 = 2³ x 5²

Conclusion

Understanding prime factorization is a fundamental concept in mathematics. By following the steps outlined above, you can confidently find the prime factorization of any number, including 200, which is definitively 2³ x 5². Remember to practice and you'll master this skill in no time!