Problem 60 - Algebra 1: Concepts and Skills

Question

Determine whether the number is prime or composite. If it is composite, give its prime factorization. $$ 99 $$

Step-by-Step Solution

Answer

The number 99 is a composite number, and its prime factorization is \(3^2 * 11\).

Step 1: Determine if the number is prime or composite

Begin by checking if the number is divisible by integers other than 1 and itself. To do so, commence from 2 and progress up to the number's square root. In our case, the square root of 99 is, to the nearest whole number, 10. We find that 99 is divisible by 3 and 33, therefore, 99 is a composite number.

Step 2: Compute the prime factorization

The goal in this step is to express 99 as a product of prime factors. This ordinarily involves dividing the number by the smallest prime, 2, and if it's not evenly divisible, proceed to the next prime, 3, and so on, until the quotient is a prime. For 99, we firstly divide it by 2. It's not divisible, thus proceed to the next prime, 3 where 99 divided by 3 is 33. Now, dividing 33 by the smallest prime, 2, is also not possible, hence proceed to the next which again is 3. We find that 33 divided by 3 is 11, and 11 is a prime number. Therefore, we can't continue the division process.

Step 3: Write down the prime factorization

Now, gather all the prime factors obtained. This incorporates the prime quotients and the primes that divided evenly into our number. Accordingly, the prime factorization of 99 is \(3 * 3 * 11\), also represented as \(3^2 * 11\).

Key Concepts

Prime Numbers

Understanding prime numbers is essential for the process of prime factorization. A prime number is a natural number greater than 1 that is not divisible by any other numbers except 1 and itself. This means:

• A prime number has exactly two distinct positive divisors: 1 and the number itself.
• The smallest prime number is 2, which is also the only even prime number.
• Examples of prime numbers include 2, 3, 5, 7, 11, and 13.

These numbers form the building blocks for other numbers because any positive integer greater than 1 can be written as a product of prime numbers. This process is known as prime factorization.

Composite Numbers

Composite numbers, on the other hand, are numbers greater than 1 that have more than two positive divisors. This means:

• Composite numbers can be divided evenly by numbers other than just 1 and themselves.
• They have at least one divisor other than 1 and the number itself.
• Examples of composite numbers include 4, 6, 8, 9, 10, and 12.

To determine if a number is composite, you can test for divisibility beginning with the smallest prime numbers. If a number can be divided evenly by any of these, it is composite. For example, 99 is a composite number because it can be divided by 3 and 33 evenly.

Prime Factors

Prime factors are the prime numbers that, when multiplied together, result in the original number. Identifying the prime factors of a number is what we call prime factorization. Here's how it works:

• Begin dividing the number by the smallest prime number, which is 2. If it divides evenly, that prime number is a factor.
• Continue the process with the quotient until you reach a quotient that is a prime number.
• If a number is not divisible by a smaller prime, move to the next largest prime number.

For instance, to find the prime factors of 99:

• Since 99 is not divisible by 2, we start with the next prime, which is 3.
• 99 divided by 3 is 33. Thus, 3 is a prime factor.
• Next, 33 divided by 3 equals 11, which is a prime number itself. Thus, another prime factor is 11.
• So, the prime factorization of 99 is expressed as \(3^2 \times 11\).

Understanding these fundamental concepts helps simplify complex numerical problems and aids in solving various math challenges like simplifying fractions and finding least common multiples.