Problem 79 - 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. $$98$$

Step-by-Step Solution

Answer

The number 98 factors into 2 and 49, where 49 is a perfect square.

Step 1: Identify Perfect Squares

First, list out perfect squares less than or equal to 98. These are 1, 4, 9, 16, 25, 36, 49, 64, and 81.

Step 2: Finding Factors

Find pairs of factors of 98 and check which pair includes a perfect square. The factors of 98 are (1, 98), (2, 49), (7, 14). In this case, the pair (2, 49) includes 49, which is a perfect square.

Step 3: Verify the Factorization

Verify that the product of 2 and 49 equals 98, which it does, confirming our factorization. Therefore, 98 can be expressed as a product of 2 and 49, where 49 is a perfect square.

Key Concepts

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. For example, 1, 4, 9, 16, 25, and so on, are perfect squares since they are

• \(1 = 1 \times 1\)
• \(4 = 2 \times 2\)
• \(9 = 3 \times 3\)
• \(16 = 4 \times 4\)

and so forth. Recognizing perfect squares is a crucial skill because it simplifies many mathematical operations.
When factoring numbers, recognizing perfect squares helps quickly identify one of the factors in a pair.
In our exercise, we considered perfect squares up to 98, namely: 1, 4, 9, 16, 25, 36, 49, 64, and 81. This is because perfect squares are
key candidates for creating a factorization that completely breaks down a given number.

Pair of Factors

Finding factor pairs is fundamental in understanding how a number can be decomposed into smaller parts. A number's factors are pairs of integers which, when multiplied, give the number itself. For instance, for 98, its factor pairs are

• (1, 98) because \(1 \times 98 = 98\)
• (2, 49) because \(2 \times 49 = 98\)
• (7, 14) because \(7 \times 14 = 98\)

Since one of these factors can often be a perfect square, identifying factor pairs is pivotal. This allows us to check if a perfect square exists within those pairs.
In our exercise, we found that among the factor pairs of 98, (2, 49) contained 49 as a perfect square. Understanding how to pair factors correctly simplifies factoring and eventually leads to simpler calculations or simplifications.

Product of Two Numbers

The expression 'product of two numbers' refers to the result obtained when two numbers are multiplied together. This simple operation is at the heart of finding factorizations. Factoring a number, such as 98, means expressing it as a product of two numbers where one or more could be special, such as a perfect square.
This way, we decomposed 98 into the product of 2 and 49, noted as \(2 \times 49 = 98\). This format is particularly useful in algebra and number theory, as it
enables us to see different perspectives of the same number.
Understanding the product of two numbers allows us to see how numbers interact and combine to form larger numbers, which can be essential in more advanced mathematical problems related to divisibility, equation solving, and optimization.