Problem 82 - 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. $$121$$

Step-by-Step Solution

Answer

The factorization of 121 into the product of two numbers with one being a perfect square is \(11 \times 11\).

Step 1: Identify Perfect Squares

Start by identifying perfect squares. Perfect squares are numbers like 1, 4, 9, 16, 25, etc. that can be expressed as the square of an integer.

Step 2: Find Perfect Square Factor

Look for the largest perfect square that can divide 121 evenly. Since 121 is a perfect square itself, we can start by checking if 121 is a perfect square. 121 is the square of 11 (i.e., \(11^2 = 121\)).

Step 3: Write the Factorization

Since 121 is already a perfect square, the factorization is trivial: \(121 = 11 \times 11\). Thus, 11 is the factor which is a perfect square itself, and there are no smaller perfect square factors.

Key Concepts

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. An intuitive way to understand perfect squares is to visualize them in terms of geometric shapes, like squares where each side has equal length. For example:

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

This pattern continues infinitely, with the series represented as: 1, 4, 9, 16, 25, 36, and so on.In the given problem, recognizing that 121 is a perfect square is key. It equals \(11 \times 11\), highlighting the fundamental importance of being able to identify perfect squares easily.

Integer Factorization

Integer factorization involves breaking down a number into a product of smaller integers. For any given number, these smaller integers are known as factors, and factoring helps us understand the building blocks of that number. Steps for Integer Factorization:

• Start by checking if the entire number is a perfect square, just like 121 in our example.
• Look for other possible smaller perfect square factors by dividing the number by known perfect squares.
• The process becomes easier by considering divisibility rules and working systematically with prime numbers, especially when numbers are larger or not obvious perfect squares.

Factors, whether a perfect square or not, can provide insights into mathematical properties of numbers, and factorization is a critical skill for solving more complex mathematical problems.

Mathematical Problem Solving

Approaching mathematical problems effectively involves using systematic techniques and strategies to find solutions. The problem given, factorizing 121, emphasizes the importance of recognizing patterns, such as perfect squares, and using them to simplify tasks.Strategies for Problem Solving Include:

• Understanding the definitions and properties of numbers, such as what constitutes a perfect square.
• Using a step-by-step approach to identify solutions, just like breaking a complex problem into smaller, manageable pieces.
• Verification of each step by checking calculations and ensuring each factor or solution is mathematically precise.

In our example, by knowing and applying properties of perfect squares and understanding integer factorization, we simplify 121 into \(11 \times 11\). It demonstrates that clear understanding, patience, and practice lead to confident and accurate problem solving in mathematics.