What Is The Prime Factorization Of 400

What is the Prime Factorization of 400? A Step-by-Step Guide

Finding the prime factorization of a number is a fundamental concept in mathematics, crucial for understanding number theory and simplifying complex calculations. This article will guide you through the process of finding the prime factorization of 400, explaining the method and highlighting key concepts. Understanding prime factorization helps with various mathematical operations, including simplifying fractions and finding the greatest common divisor (GCD) or least common multiple (LCM) of numbers.

What is Prime Factorization?

Prime factorization is the process of expressing a composite number (a number greater than 1 that is not prime) as a product of its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 2, 3, 5, 7, 11 are prime numbers. The prime factorization of a number is unique; meaning there's only one way to express it as a product of prime numbers (ignoring the order).

Finding the Prime Factorization of 400

We can use a factor tree or repeated division to find the prime factorization of 400. Let's use the factor tree method:

1. Start with the number 400: We look for the smallest prime number that divides 400 evenly. That's 2.

2. Divide by 2: 400 / 2 = 200. We now have 2 x 200.

3. Continue dividing by 2: 200 / 2 = 100. Now we have 2 x 2 x 100.

4. Repeat the process: 100 / 2 = 50. This gives us 2 x 2 x 2 x 50.

5. Keep going: 50 / 2 = 25. We now have 2 x 2 x 2 x 2 x 25.

6. Find the next prime factor: 25 is not divisible by 2, but it is divisible by 5 (another prime number).

7. Divide by 5: 25 / 5 = 5. We now have 2 x 2 x 2 x 2 x 5 x 5.

8. The last prime factor: 5 is a prime number.

Therefore, the prime factorization of 400 is 2 x 2 x 2 x 2 x 5 x 5, which can be written more concisely as 2⁴ x 5².

Understanding the Result

The prime factorization tells us that 400 is composed of four 2's and two 5's multiplied together. This representation is useful for various mathematical operations. For instance, finding the square root of 400 becomes easier: √400 = √(2⁴ x 5²) = 2² x 5 = 20.

Alternative Method: Repeated Division

Alternatively, you can use repeated division. Continuously divide the number by the smallest prime number possible until you reach 1.

400 ÷ 2 = 200
200 ÷ 2 = 100
100 ÷ 2 = 50
50 ÷ 2 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1

The prime factors are the divisors used: 2, 2, 2, 2, 5, 5. This confirms our previous result of 2⁴ x 5².

This comprehensive guide demonstrates how to find the prime factorization of 400 using both the factor tree method and repeated division. Understanding this fundamental concept is crucial for further exploration in number theory and various mathematical applications.