Problem 22 - Introductory Algebra for College Students

Question

Identify each natural number as prime or composite. If the number is composite, find its prime factorization. $$110$$

Step-by-Step Solution

Answer

110 is a composite number. Its prime factorization is \(2 \times 5 \times 11\).

Step 1: Identify Prime or Composite

We begin by assessing if the number has more than two distinct positive divisors. If we can find any number other than 1 and 110 that divides evenly into 110, it will prove to be composite. A quick check shows that 110 is divisible by 2, 5, 10, 11, 22, 55, proving it is a composite number.

Step 2: Prime Factorization

Given that 110 is a composite number, we proceed to find its prime factorization. We divide the number by prime numbers, starting from the smallest (2, 3, 5, 7, 11, 13,...) and continue until we cannot divide anymore. For 110, we see that it is divisible by 2, yielding 2 and 55. We continue to divide 55 by the smallest prime number, but 55 isn't divisible by 2, 3, and 5. So we divide it by the next prime number which is 5. We get 5 and 11 after dividing. 11 further can't be divided as it is a prime number. Always remember that prime numbers can't be divided to get other factors. So, the prime factors of 110 are 2, 5, and 11.