Problem 25 - Introductory Algebra for College Students

Question

Identify each natural number as prime or composite. If the number is composite, find its prime factorization. $$81$$

Step-by-Step Solution

Answer

The number 81 is a composite number. Its prime factorization is \(81 = 3^4\).

Step 1: Identify the Number as Prime or Composite

To find if the number 81 is prime or composite, check if the number has any other divisors apart from 1 and itself. In this case, 81 can be divided evenly by other numbers, such as 3 and 27, so it is identified as a composite number.

Step 2: Find the Prime Factorization

To find the prime factorization of 81, we should divide the number by prime numbers, starting from the smallest (2), and continue until the result is a prime number. However, 81 is odd, so it can't be divided by 2. So, moving on to the next smallest prime number, 3, 81 is divided by 3 to give 27. Again, 27 can be divided by 3 to give 9, and subsequently 9 divided by 3 gives 3. So, \(81 = 3 × 3 × 3 × 3\), or, in general terms, also known as \(81 = 3^4\). Here, we see that 4 is the power of the prime number 3, indicates that 3 is used 4 times to multiply to get the composite number 81.