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Chapter 10: Problem 436

Find the greatest common factor. $$24 u, 6 u^{2}, 30 u^{3}$$

Short Answer

Expert verified
The greatest common factor is 6u.

Step by step solution

01

List factors of each coefficient

Identify the factors of each coefficient part of the given terms. For 24, the factors are 1, 2, 3, 4, 6, 8, 12, 24. For 6, the factors are 1, 2, 3, 6. For 30, the factors are 1, 2, 3, 5, 6, 10, 15, 30.
02

Find the common factors

Observe the common factors among 24, 6, and 30. The common factors are 1, 2, 3, and 6.
03

Determine the greatest common factor

Among the common factors (1, 2, 3, 6), the greatest is 6.
04

Identify the lowest power of the common variable

The common variable in all terms is u. The lowest power of u among the terms is u.
05

Combine the greatest common factor and the lowest power of the variable

Combine the greatest common factor 6 and the lowest power of the variable u to find the greatest common factor, which is 6u.

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Key Concepts

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

Factors
To understand the concept of the greatest common factor (GCF), we first need to know what factors are. Factors are numbers that you can multiply together to get another number. For example, the factors of 24 are all the numbers that can divide 24 without leaving a remainder. These numbers are 1, 2, 3, 4, 6, 8, 12, and 24.

When listing factors of a number, always start with 1 and the number itself. Continue by testing smaller numbers to see if they divide evenly into the number. This process helps in identifying all the factors easily.
Common Factors
Common factors are factors that two or more numbers have in common. To find the common factors of numbers, you must list out the factors of each number and then find the numbers that appear in all lists. For example, in the exercise, the factors of 24 (1, 2, 3, 4, 6, 8, 12, 24), 6 (1, 2, 3, 6), and 30 (1, 2, 3, 5, 6, 10, 15, 30) show that the numbers 1, 2, 3, and 6 are common factors.

From these common factors, you then identify the greatest one. In this example, 6 is the greatest number that appears in all lists, so it's the greatest common factor.
Variables
Variables are symbols used to represent numbers or values in mathematical expressions. The most common variable is usually 'x,' but in the context of the given exercise, 'u' is used. When considering variables in the context of factors, you must look for the lowest power of the common variable.

For example, in the terms \(24u\), \(6u^2\), and \(30u^3\), the common variable is 'u'. The term \(24u\) has the variable 'u' raised to the first power, which is the lowest power among all the terms. You must then combine this lowest power of the variable with the greatest common factor of the numerical coefficients (6) to get the final greatest common factor, \(6u\).

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Most popular questions from this chapter

Multiply and write your answer in decimal form. $$\left(3 \times 10^{7}\right)\left(2 \times 10^{-4}\right)$$

Simplify. $$6^{-2}$$

Multiply and write your answer in decimal form. $$\frac{6 \times 10^{9}}{2 \times 10^{-1}}$$

Find the greatest common factor. $$12 x^{2}, 20 x^{3}, 36 x^{4}$$

Simplify each expression. $$\left(8 x y^{3}\right)\left(-6 x^{4} y^{6}\right)$$
See all solutions

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