Find the Prime Factorization of 32, using the Factor Tree.
The Correct Answer and Explanation is:
Prime Factorization of 32 Using a Factor Tree
To find the prime factorization of 32 using a factor tree, we begin by breaking it down into smaller factors until all the numbers are prime.
Step-by-step factor tree:
- Start with 32
- 32 is divisible by 2 (the smallest prime number):
32 = 2 × 16 - Now factor 16:
16 = 2 × 8 - Next, factor 8:
8 = 2 × 4 - Factor 4:
4 = 2 × 2
Now all the branches of the tree end in the number 2, which is a prime number. So the prime factorization of 32 is:
2 × 2 × 2 × 2 × 2 or 2⁵
Explanation
Prime factorization is the process of breaking down a composite number into the set of prime numbers that multiply together to give the original number. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, and 13. The number 2 is particularly important because it is the smallest and only even prime number.
In the case of 32, we start by recognizing that it is an even number. Since all even numbers are divisible by 2, we divide 32 by 2 to get 16. We continue this process with the result of each division. Dividing 16 by 2 gives 8, dividing 8 by 2 gives 4, and dividing 4 by 2 gives 2. At this point, we reach 2 again, and since dividing 2 by 2 gives 1, we know we have reached the end of our factor tree. The number 1 is not included in the prime factorization because it is not a prime number.
Each step in this process involves dividing by the smallest possible prime number until only prime numbers remain. In this case, every factor was 2. Therefore, the prime factorization of 32 is five twos multiplied together, or 2 × 2 × 2 × 2 × 2, which can be written in exponential form as 2⁵.
This method not only helps in understanding the structure of numbers but also plays a crucial role in simplifying fractions, finding greatest common divisors, and solving many math problems involving multiples and factors.
