Problem 94 - Algebra 1: Concepts and Skills

Question

Write the prime factorization of the number if it is not a prime number. If a number is prime, write prime. $$ 96 $$

Step-by-Step Solution

Answer

The prime factorization of 96 is \(2^{5} × 3^{1}\).

Step 1: Determine if the Number is Prime

In order to find out if a number is prime, all of its factors need to be checked. A prime number only has two factors - 1 and itself. The given number is 96. Checking quickly, it is obvious that 96 is not a prime number as it's even and can therefore be divided by 2 at least. Hence, 96 is not a prime number.

Step 2: Perform Prime Factorization

The factorization process includes dividing the given number starting from the smallest prime number (which is 2) and continue dividing until the quotient is a prime number. Start by dividing 96 by 2: \(96 ÷ 2 = 48\). The quotient 48 is not a prime number, so continue dividing by 2: \(48 ÷ 2 = 24\). Next, divide 24 by 2: \(24 ÷ 2 = 12\), then divide 12 by 2: \( 12 ÷ 2 = 6 \), and finally divide 6 by 2: \( 6 ÷ 2 = 3 \). Now, 3 is a prime number, so the process stops here.

Step 3: Write the Prime Factorization

The prime factorization of a number is all the prime numbers that multiply together to give the original number. From the steps above, the prime factors of 96 are all the divisors used to get to the number 3. So the prime factorization of 96 is written as: \(2 × 2 × 2 × 2 × 2 × 3\), or in exponent form \(2^{5} × 3^{1}\). This means that 96 can be expressed as the product of five 2s and one 3.

Key Concepts

Factors

When we talk about factors, we are talking about numbers that can multiply together to make another number. For instance, if you can multiply two numbers together to get a third number, then those two numbers are factors of that third number.
In the context of the number 96, factors are numbers that can be multiplied together to equal 96. For example:

• 1 and 96 are factors of 96, because 1 × 96 = 96.
• 2 and 48 are also factors, because 2 × 48 = 96.

Factors come in pairs. They can be small or large, depending on the number you start with.
Understanding what factors are helps us look for prime factors, which are crucial in prime factorization.

Prime Number

A prime number is a special kind of number that has exactly two distinct factors: 1 and itself. This means that a prime number cannot be divided by any other number without leaving a remainder.
Some examples of prime numbers are 2, 3, 5, 7, and 11. All these numbers only divide evenly by 1 and themselves.
On the other hand, any number that has more than two factors isn't a prime number. For example, as noted in the exercise, 96 is not a prime number. This is because 96 can be divided evenly by numbers other than 1 and itself, such as 2, 3, 4, etc.
Recognizing whether a number is prime or not is crucial because only prime numbers go into the prime factorization process.

Exponent Form

Once you've determined that a number is not prime and have found its prime factors, you can use exponent form to simplify how you represent these factors.
Exponent form is a shorthand way of writing repeated multiplication of the same number. For example, rather than writing 2 × 2 × 2 × 2 × 2, we can write it as 2 raised to the power of 5, which is written as \(2^{5}\).
This gives us a more compact and tidy way to express prime factorization. As such, the prime factorization of 96, which is 2 × 2 × 2 × 2 × 2 × 3, can be neatly written in exponent form as \(2^{5} \times 3^{1}\).
The use of exponents is helpful in many areas of math because it highlights how many times a specific factor is used.