Problem 78 - Prealgebra

Question

Factor each of the following numbers into the product of two numbers, one of which is a perfect square. (Remember from Chapter 1, a perfect square is \(1,4,9,16,25,36, \ldots\) etc. $$27$$

Step-by-Step Solution

Answer

27 can be factored as 9 (a perfect square) times 3.

Step 1: Identify the perfect squares less than or equal to the number

Start by identifying all perfect squares less than 27. The perfect squares are: 1, 4, 9, 16, and 25.

Step 2: Find a perfect square as a factor

Among the perfect squares, check which ones can divide 27 without leaving a remainder. Test each one by division.

Step 3: Test division with 9

Divide 27 by 9 since 9 is a perfect square: \[ \frac{27}{9} = 3 \]9 times 3 equals 27, so it works.

Step 4: State the factorization

Factor 27 into the product of a perfect square and another number as follows: 27 equals 9 times 3. This can be written as:\[ 27 = 9 \times 3 \]

Key Concepts

Perfect Squares

A perfect square is a number that is the square of an integer. Understanding perfect squares is essential when breaking down numbers, as they provide a useful starting point for factoring. Common perfect squares you might recognize include:

• 4, which equals \( 2^2 \)
• 9, which equals \( 3^2 \)
• 16, which equals \( 4^2 \)
• 25, which equals \( 5^2 \)
• 36, which equals \( 6^2 \)

Identifying perfect squares within a number helps you to simplify problems more efficiently. You start by listing down perfect squares that are less than or equal to the given number. This list acts as a toolkit for identifying factors that can assist in simplifying your calculations. The next step in this process is using this list to determine which of these perfect squares can divide your number without leaving a remainder. This leads us into our next topic: division.

Division

Division in this context refers to the act of determining which perfect squares can evenly divide the number you are trying to factor. It means dividing the number by each potential perfect square from your list until you find one that leaves no remainder. For example, if you're factoring the number 27, you would start by dividing it by each perfect square:

• 27 divided by 1 equals 27. This means 1 is a factor, but it's not very helpful initially.
• 27 divided by 4 equals 6.75, indicating that 4 is not a factor because the result is not whole.
• 27 divided by 9 equals 3. Now the equation balances with no remainder, which means 9 is a factor of 27.

Through division, you identify which perfect square (in this case, 9) can be part of the factorization problem's solution. Once this dividing step is completed and verified, you have confirmed a perfect square is a factor. This is an important bridge to understanding how **prime factorization** is achieved.

Prime Factorization

Prime factorization involves breaking down a number into its smallest prime components, and it often incorporates perfect squares as a step in the process. After identifying a perfect square as a factor, prime factorization allows for a deeper breakdown. In the example with 27, once you divide it successfully by the perfect square 9, the resulting factor is 3.

• 9 itself can also be broken down into prime factors, as 3 times 3 (since 9 is \(3^2\)).
• Combining, this results in a full factorization of 27 being \(3^3\).

Why is this important? Prime factorization reveals all the basic building blocks of a number, which can then be reassembled or recombined into different factorizations. In practical terms, knowing a number’s prime factors can help with other mathematical operations and provide insight into its structure. This knowledge sets a solid foundation for solving a variety of mathematical problems effortlessly.