Problem 75 - 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. $$40$$

Step-by-Step Solution

Answer

Factor 40 as 4 (perfect square) and 10.

Step 1: Identify Perfect Squares

First, we need to determine which perfect squares could potentially be factors of 40. The perfect squares less than 40 are 1, 4, 9, 16, and 25.

Step 2: Check for Factors

Check if 40 can be divided evenly by any of these perfect squares. Start with the largest perfect square. 40 divided by 25 is not an integer, similarly, neither is 40 divided by 16. But 40 divided by 4 equals 10, which is an integer.

Step 3: Verify the Factor Pair

This means that 4 is a factor of 40, with 10 being the co-factor. Since 4 is a perfect square, the factor pair of 40 that includes a perfect square is (4, 10).

Key Concepts

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. In other words, it's the result of multiplying an integer by itself. For example:

• 1 is a perfect square because it's 1 times 1.
• 4 is a perfect square because it's 2 times 2.
• 9 is a perfect square because it's 3 times 3.

Perfect squares play a crucial role in factoring because they simplify equations and make calculations more manageable.
A list of the first few perfect squares includes numbers like 1, 4, 9, 16, 25, 36, and so on. In the exercise we're examining, 40 was factored using 4, which is one of these perfect squares. Recognizing perfect squares in a number helps identify a structure and predictability, making problem-solving smoother.

Integers

Integers are whole numbers that can be positive, negative, or zero, and they do not have fractional or decimal parts. Some examples are -3, 0, 7, and 12. When factoring numbers, particularly involving perfect squares, we often look for integer solutions.
When examining the factorization of an integer like 40, you determine which numbers divide it evenly, resulting in another integer. In our example, 40 divided by the perfect square 4 gives us 10, another integer, indicating that 4 is indeed a factor.
Understanding how integers work is fundamental in math because it helps in determining factor pairs, identifying divisibility, and ensuring that solutions are sensible and practical.

Co-factor

A co-factor is a number that works alongside another number to form a product, often in the context of factorization. When you factor a number into a perfect square and another integer, the integer is typically referred to as the co-factor of that perfect square.
Using our example with 40, once we identified 4 as a perfect square factor, we found that 10 is the co-factor because multiplying 4 and 10 gives us 40.
Working with co-factors helps in understanding the structure of numbers and recognizing their divisible components. Finding a co-factor is key in breaking down numbers into more manageable pieces, which is particularly helpful in algebra and number theory.