Problem 28 - Elementary and Intermediate Algebra

Question

Find the prime factorization of each number. If the number is prime, state this. $$ 54 $$

Step-by-Step Solution

Answer

54 = 2 × 3³.

Step 1: Identify if the Number is Prime

Determine if 54 is a prime number. Since it is not greater than 1 and cannot only be divided by 1 and itself, it is not a prime number.

Step 2: Start with the Smallest Prime Number

Begin dividing 54 by the smallest prime number, which is 2. 54 ÷ 2 = 27. This means 54 can be factored as 2 × 27.

Step 3: Continue Factoring 27

Now, take 27. Since it is not divisible by 2, try the next smallest prime number, which is 3. 27 ÷ 3 = 9. So now we have the factorization: 54 = 2 × 3 × 9.

Step 4: Factor Further if Possible

Next, take 9. Since it is not divisible by 2, use the prime number 3 again. 9 ÷ 3 = 3, giving us: 54 = 2 × 3 × 3 × 3.

Step 5: Confirm All Numbers are Prime

Now confirm that all the factors (2, 3, 3, 3) are prime numbers. Since each factor is prime, the prime factorization of 54 is complete.

Key Concepts

prime numbers

Prime numbers are the building blocks of whole numbers. They are greater than 1 and can only be divided by 1 and themselves without leaving a remainder. For example, 2, 3, 5, and 7 are prime numbers. These special numbers play a crucial role in number theory and are used in various fields like cryptography and computer science.

factorization

Factorization is the process of breaking down a number into smaller numbers that can be multiplied together to give the original number. Think of it like breaking a big problem into smaller, more manageable pieces. For instance, if you have the number 54, you can factor it into 2 × 27 or 2 × 3 × 9, until you get all the smallest building blocks, like 2 × 3 × 3 × 3.

division method

The division method helps in finding the prime factors of a number by dividing it repeatedly by the smallest prime numbers. Here's how you can use it:
• Start with the smallest prime number, which is 2.
• Divide your number (in this case, 54) by 2, and if it divides evenly, write down 2 as a factor and keep the quotient.
• Continue dividing the quotient by 2 until it no longer divides evenly.
• Move to the next smallest prime number (3) and repeat.
• Continue this process until all the factors are prime numbers.
By dividing step-by-step, you ensure that you methodically break down the number into its simplest prime factors.

prime factors

Prime factors are the prime numbers that you multiply together to get the original number. In our example, 54, the prime factors are 2, 3, 3, and 3. So, the prime factorization of 54 is 2 × 3 × 3 × 3. Each of these factors is a prime number, and when multiplied together, they give you 54. Knowing the prime factors of a number is very useful in many areas of mathematics and science.