Problem 71 - 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. $$75$$

Step-by-Step Solution

Answer

75 = 25 \times 3, where 25 is a perfect square.

Step 1: Identifying Perfect Squares

Understand that a perfect square is a number that can be expressed as the square of an integer, such as 1, 4, 9, 16, etc. We need one factor to be a perfect square.

Step 2: Finding Factors of 75

List the factors of 75 to determine potential perfect squares. The factor pairs are (1, 75), (3, 25), (5, 15), and (15, 5).

Key Concepts

Perfect Squares

A perfect square is a number that can be represented as the product of an integer multiplied by itself. For example, when you multiply 3 by itself, you get 9; thus, 9 is a perfect square. Recognizing these numbers can simplify complex calculations and enable easier factorization of larger numbers.
Common examples of perfect squares include:

• 1 (since 1 x 1 = 1)
• 4 (since 2 x 2 = 4)
• 9 (since 3 x 3 = 9)
• 16 (since 4 x 4 = 16)
• 25 (since 5 x 5 = 25)
• 36, 49, and so on...

Spotting perfect squares is crucial when it comes to factoring numbers, as it greatly reduces the number of possible combinations you need to examine, especially when dealing with large numbers.
To determine if a number is a perfect square, simply check if it can be expressed as the square of an integer, or in mathematical terms, if it has an integer square root.

Integer Factorization

Integer factorization involves breaking down a larger number into a product of smaller integers, specifically prime numbers or perfect squares. It's a way of decomposing a number into a set of factors that, when multiplied together, will give the original number.
Take the number 75, for example. To factorize this number, consider possible integers that multiply together to get 75. The factors of 75 include: 1, 3, 5, 15, 25, and 75.
Using integer factorization:

• We understand that 75 can be decomposed into (3 x 25).
• Notice here that 25 is a perfect square (5 x 5).

Therefore, the integer factorization of 75 emphasizes the importance of identifying perfect squares because it allows us to quickly simplify and express the number as a product of familiar terms.

Mathematical Factors

Mathematical factors are numbers you multiply together to get another number. They play an essential role in understanding various mathematical propositions and solving problems. For example, when you factor a number, you essentially break it into components that are easier to handle and understand.
Let's look back at the number 75. To find its factors:

• You can start by dividing it by the smallest prime number, which is 3, to get 25.
• Next, recognizing that 25 is a perfect square, you can express 75 as (3 x 25).
• There’s also the factor pair (5 x 15), which involves multiplying smaller integers.

By factoring, you simplify a number, potentially revealing properties that are not immediately obvious. Having a solid grasp of how to find these factors aids greatly in problem-solving and mathematical understanding.