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Chapter 10: Problem 1

Which is the greatest perfect square factor of \(128 ?\) A. 12 B. 16 C. 32 D. 64

Short Answer

Expert verified
The greatest perfect square factor of 128 is 64.

Step by step solution

01

- Identify the Factors

Determine all factors of 128. Factors are numbers that divide 128 without leaving a remainder. The factors of 128 are: 1, 2, 4, 8, 16, 32, 64, and 128.
02

- Identify the Perfect Squares

Out of the factors of 128, identify which ones are perfect squares. Perfect squares are numbers that can be expressed as the square of an integer. The perfect squares in the factors of 128 are: 1 (1^2), 4 (2^2), and 16 (4^2), and 64 (8^2).
03

- Identify the Greatest Perfect Square

Compare the perfect square factors to determine the greatest one. Among 1, 4, 16, and 64, the greatest perfect square is 64.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is like breaking something down into its smaller pieces.
For example, when you factor a number, you are finding all the numbers that multiply together to make the original number.
In this exercise, we began by finding all the factors of 128.
This means identifying all numbers that can be multiplied together to give 128 without leaving a remainder.
Here, the factors of 128 are: 1, 2, 4, 8, 16, 32, 64, and 128.
Remember, when you find the factors of a number, you are not just listing numbers but ensuring that these numbers evenly divide the original number.
Perfect Squares
A perfect square results from squaring an integer.
This means multiplying an integer by itself.
For instance, \({2^2 = 4}\) and \({8^2 = 64}\) where both 4 and 64 are perfect squares.
In this exercise, among the factors of 128, we identified which ones were perfect squares.
The perfect squares from the factors are: 1 (1虏), 4 (2虏), 16 (4虏), and 64 (8虏).
Knowing perfect squares helps in many areas of mathematics, especially in solving equations and understanding number properties.
Greatest Common Factor
The Greatest Common Factor (GCF), also known as the greatest common divisor, is the largest number that can exactly divide two or more numbers.
In this exercise, while we didn't need to find the GCF of multiple numbers, we looked for the greatest perfect square factor of 128.
Out of the factors of 128 that were also perfect squares, we had 1, 4, 16, and 64.
By comparing these perfect squares, 64 is the greatest.
This demonstrates how understanding and identifying the GCF can help in various mathematical problems.
Knowing the GCF can simplify fractions, reduce equations, and solve problems involving ratios.
Integer Properties
Integers are whole numbers which include positive, negative numbers, and zero.
Properties of integers include understanding how they interact through various mathematical operations.
For this exercise, understanding properties of integers such as factors and perfect squares helped in breaking down the problem.
All the numbers listed as factors of 128 were integers.
Additionally, the perfect squares we identified (1, 4, 16, and 64) are also integers.
Understanding integer properties is crucial for performing accurate calculations and solving complex mathematical problems efficiently.

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