Problem 28 - Introductory Algebra for College Students

Question

Identify each natural number as prime or composite. If the number is composite, find its prime factorization. $$360$$

Step-by-Step Solution

Answer

360 is a composite number and its prime factorization is \(2^3 \times 3^2 \times 5\)

Step 1: Identify the Number Type

To check whether 360 is a prime or composite number, try to find any divisors other than 1 and itself. Since 360 is divisible by numbers such as 2, 3, 5, and others, we can conclude that 360 is a composite number.

Step 2: Initial Prime Factorization

To find the prime factorization of 360, start by dividing it by the smallest prime number, which is 2. 360 divided by 2 is equal to 180.

Step 3: Continued Prime Factorization

Continue dividing by 2 until it's no longer divisible evenly. So 180 is divisible by 2 and gives 90, 90 is divisible by 2 and gives 45. The number 45 is no longer divisible by 2, so proceed to the next prime number which is 3. Divide 45 by 3, to get 15. Then, divide 15 by 3, to get 5. Since 5 is a prime number and is not divisible by any other prime number, the process ends here.

Step 4: Construct the Prime Factorization

Collect all the prime factors obtained from the division process. These are 2, 2, 2, 3, 3 and 5. This means that the prime factorization of 360 is \(2^3 \times 3^2 \times 5 \)