Problem 80 - 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. $$72$$

Step-by-Step Solution

Answer

The factorization of 72 into a product of a perfect square is 36 and 2.

Step 1: Identify Perfect Squares Less Than 72

List the perfect squares less than 72 to consider them for being one of the factors. These are 1, 4, 9, 16, 25, 36.

Step 2: Try Dividing by Largest Perfect Square

Start with 36, which is the largest perfect square less than 72. Divide 72 by 36. Since 72 ÷ 36 = 2, both 36 and 2 are factors, and 36 is a perfect square.

Step 3: Check the Calculation

Verify the calculation by multiplying the factors. Compute 36 × 2 = 72 to confirm the factorization is correct.

Key Concepts

Perfect Squares

Perfect squares are numbers that are the product of an integer multiplied by itself. For instance, if you take 3 and multiply it by 3, you get 9, which is a perfect square. In this exercise, recognizing perfect squares is important because they play a crucial role in our factorization task.

Here is a quick list of common perfect squares:

• 1
• 4
• 9
• 16
• 25
• 36

Understanding perfect squares helps in simplifying the factorization process by allowing us to identify one of the desired factors right away.

Factors

Factors of a number are whole numbers that multiply together to produce the original number. For instance, the factors of 72 include numbers like 1, 2, 4, 6, 9, 12, 18, 24, 36, and 72 itself. We have an endless possibility of pairings, but our goal in this particular exercise is to find a pair where one number is a perfect square, like 36.
To find factors, you can list numbers in a systematic way, thereby ensuring you cover all possibilities. Start from 1 and the number itself, and work inward.
Factors are extremely handy in problem-solving because they reveal a lot about the number’s structure and simplify calculations.

Mathematics Problem-Solving

Mathematics problem-solving involves a step-by-step approach to finding a solution. In this exercise, we break down the task into manageable steps using logical operations. This systematic approach helps in avoiding errors and reaching the correct answer efficiently.

Key steps can include:

• Identifying known quantities (like perfect squares)
• Applying arithmetic operations (such as division)
• Verifying calculations

Each part of the process is essential for confirming that the solution is accurate and satisfies the requirements of the problem at hand.

Arithmetic Operations

Arithmetic operations are basic calculations that include addition, subtraction, multiplication, and division. In our factorization exercise, division and multiplication are prominently used to separate and verify factors.
When we divided 72 by 36, it specifically told us that these two numbers work as factors. Multiplication was used next to check if multiplying these factors returned to the original number, which is essential to confirm their correctness.

Familiarity with these operations allows for smooth progress through numerical challenges and helps in not losing track of the objective while working through the calculations.