Problem 76 - 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. $$18$$

Step-by-Step Solution

Answer

18 can be factored into 9 and 2, where 9 is a perfect square.

Step 1: Identify Perfect Squares

First, list the perfect squares up to the number 18. These include 1, 4, 9, and 16.

Step 2: Find Divisors That Are Perfect Squares

Examine the divisors of 18 to see which are perfect squares. The divisors of 18 are 1, 2, 3, 6, 9, and 18. Out of these, 1 and 9 are perfect squares.

Step 3: Form Products

Pair 18 with each perfect square divisor to form a product. This gives 18 = 1 × 18 and 18 = 9 × 2.

Step 4: Select Viable Solution

Among pairs, 9 × 2 satisfies the condition where one of the numbers in the product is a perfect square (9).

Key Concepts

Perfect Squares

Perfect squares are numbers that are the square of an integer. This means that you get a perfect square when you multiply an integer by itself. It's like having a number that fits perfectly in both length and width, like a square-shaped tile. For example, the perfect squares less than 18 are:

• 1 (since 1 x 1 = 1)
• 4 (since 2 x 2 = 4)
• 9 (since 3 x 3 = 9)
• 16 (since 4 x 4 = 16)

Understanding perfect squares can help you break down numbers into factors and identify efficient ways to solve problems. Keep in mind, not all numbers are perfect squares, but they often appear in math problems like factoring.

Number Divisors

A divisor of a number is a value that can be divided by the number without leaving a remainder. Divisors play a crucial role in factoring exercises, where we try to express a number as a product of smaller numbers. When you have a number like 18, you look for numbers that it can be evenly divided by:

• 1 is a divisor of every number.
• 2 is a divisor since 18 divided by 2 is 9.
• 3 is a divisor since 18 divided by 3 is 6.
• 6 is a divisor since 18 divided by 6 is 3.
• 9 is a divisor since 18 divided by 9 is 2.
• 18 is a divisor of itself.

In our original exercise, we focus on divisors of 18 that are also perfect squares, such as 1 and 9. Recognizing divisors helps in breaking down numbers systematically.

Products of Numbers

The process of factoring involves expressing a number as a product of other numbers, much like breaking it down into puzzle pieces that fit together to recreate the number. For example, 18 can be written as:

• 1 x 18
• 9 x 2

In these expressions, 9 x 2 is special because 9 is a perfect square. This method helps us understand the makeup of numbers by using their divisors. When breaking things down like this, having one factor as a perfect square can simplify problems and lead to more intuitive solutions. Remember, factoring is not just about finding any two numbers; sometimes the task will require one of those numbers to be more specific, like a perfect square, to fulfill the conditions of the math problem.