Problem 87 - Algebra A Combined Function

Question

Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as \(9 \cdot 2,\) because 9 is a perfect square. $$ 75 $$

Step-by-Step Solution

Answer

75 can be expressed as a product of 25 (a perfect square) and 3.

Step 1: Identify Perfect Squares

Determine the perfect squares less than or equal to 75. These are 1, 4, 9, 16, 25, 36, and 49. However, only those that can be factors of 75 will be of interest.

Step 2: Identify Factor Pairs

List the factor pairs of 75. These include: (1, 75), (3, 25), (5, 15), and (9, ~8.33). We can ignore any factor pair where both factors are not integers.

Step 3: Find Suitable Factor

Among the factor pairs identified, the pair (3, 25) includes 25 as one of its factors. Since 25 is a perfect square ( 25 = 5^2 ), it is suitable for our exercise.

Step 4: Write Integer as Product

Write 75 as the product of its factors, one of which is a perfect square: \(75 = 25 \times 3\), where 25 is a perfect square.

Key Concepts

Perfect Squares

A perfect square is a number that is the square of an integer. This means it is the product of an integer multiplied by itself. For example, numbers like 1, 4, 9, 16, and 25 are perfect squares. To find perfect squares, you simply multiply a number by itself. Here are some examples:

• 1 is a perfect square because it is \(1 \times 1 = 1\).
• 4 is a perfect square because it is \(2 \times 2 = 4\).
• 9 is a perfect square because it is \(3 \times 3 = 9\).
• 16 is a perfect square because it is \(4 \times 4 = 16\).
• 25 is a perfect square because it is \(5 \times 5 = 25\).

Understanding perfect squares helps in solving many mathematical problems, including the one addressed in this exercise.

Multiplication

Multiplication is one of the basic operations in mathematics. It involves finding the product of two numbers, known as factors. When you multiply two numbers, you are essentially adding one of the numbers to itself, repeatedly, the number of times indicated by the other number. For example:

• When you multiply 3 and 25, you are doing \(3 + 3 + 3 + \ldots\) (25 times), or \(25 + 25 + 25\).
• You get that \(3 \times 25 = 75\), which fits our exercise problem very well.

Multiplication is crucial in various areas of mathematics expanding beyond simple arithmetic to more complex algebraic expressions. It's a simple yet powerful tool for expressing larger sums in a compact form.

Factor Pairs

Factor pairs are sets of two numbers that multiply together to produce another number. In this exercise, we're interested in factor pairs of 75. Factor pairs can help you break down a number into building blocks, showing how multiplication leads to the product. To find factor pairs:

• Determine which numbers can be multiplied to equal the target number, here it is 75.
• Examples of factor pairs for 75 are: (1, 75), (3, 25), and (5, 15).

Notice that some factors might be perfect squares, like 25 in the pair (3, 25). Identifying factor pairs is a helpful skill that simplifies multiplication and helps in recognizing patterns, such as divisibility and perfect squares.

Mathematics Education

Mathematics education focuses on developing fundamental skills such as problem-solving, logical reasoning, and critical thinking. It supports the understanding of key concepts like numbers, operations, and their relations. When dealing with exercises involving integer factorization and perfect squares:

• Students learn to identify important components such as perfect squares, which simplifies complex problems.
• They gain practice in multiplication, enhancing their arithmetic skills and number sense.
• Understanding factor pairs encourages systematic thinking and helps students develop strategies for finding solutions efficiently.

Overall, mathematics education empowers students to approach abstract concepts with confidence, equipping them with tools to tackle real-world problems as well as academic challenges.