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Chapter 6: Problem 3

Find the greatest common factor for each list of numbers. \(18,24,36,48\)

Short Answer

Expert verified
The greatest common factor is 6.

Step by step solution

01

- Find the Prime Factors

Identify the prime factors of each number. For 18: 18 = 2 脳 3^2 For 24: 24 = 2^3 脳 3 For 36: 36 = 2^2 脳 3^2 For 48: 48 = 2^4 脳 3
02

- Identify the Common Prime Factors

Look for the common prime factors in all the numbers. The common prime factors are 2 and 3.
03

- Choose the Lowest Power of Each Common Prime Factor

From each set of prime factors, choose the lowest power of each common prime factor. Here, the lowest power for 2 is 2 (as 18 has 2^1), and for 3 it is 1 (as 48 has 3^1).
04

- Multiply the Lowest Powers Together

Multiply the lowest powers of each common prime factor together. The greatest common factor (GCF) is 2^1 脳 3^1 = 2 脳 3 = 6.

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Key Concepts

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

prime factors
Prime factors are the building blocks of a number. To find them, we break the number down into smaller numbers, specifically prime numbers. Remember, a prime number is a number greater than 1 that has no divisors other than 1 and itself.

For example, let's find the prime factors of 18:
  • We start with 2 (the smallest prime number) which divides 18 evenly. We get 18 梅 2 = 9.
  • Next, 9 is divided by 3 (the next smallest prime number). We get 9 梅 3 = 3.
  • Finally, 3 divided by 3 gives 1.
So, the prime factors of 18 are 2 and 3.
We write this as: 18 = 2 脳 32.

Finding prime factors for the other numbers follows the same process. For 24: 24 = 23 脳 3, for 36: 36 = 22 脳 32, and for 48: 48 = 24 脳 3.
common prime factors
After finding the prime factors of each number, the next step is to identify the common prime factors. This means finding the prime numbers that appear in the factorization of each number.

From our prime factorizations:
  • 18: 2 脳 32
  • 24: 23 脳 3
  • 36: 22 脳 32
  • 48: 24 脳 3
We notice that both 2 and 3 are present in all the factorizations.
Therefore, our common prime factors are 2 and 3. Identifying these common factors is crucial in finding the greatest common factor, or GCF, as they will be used in the next step.
lowest power
With the common prime factors identified, we now focus on the lowest power of each common prime factor. For each factor, we choose the smallest exponent (or power) that appears in the factorizations.

Let's consider the common prime factors 2 and 3:
  • For 2, the exponents are: 1 (from 18), 3 (from 24), 2 (from 36), and 4 (from 48). The lowest exponent here is 1.
  • For 3, the exponents are: 2 (from 18), 1 (from 24), 2 (from 36), and 1 (from 48). The lowest exponent here is 1.
The lowest power for each common prime factor is crucial as it determines the smallest set of multiples common to all the numbers. Multiplying these lowest powers together gives us our greatest common factor (GCF).

So, we multiply the lowest powers of each common prime factor: 21 脳 31 = 2 脳 3 = 6. Therefore, the GCF of 18, 24, 36, and 48 is 6.

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Most popular questions from this chapter

Apply the special factoring nules of this section to factor each binomial or trinomial. $$ p^{2}-\frac{1}{9} $$

Factor each trinomial completely. $$ 25 z^{4}+5 z^{3}+z^{2} $$

Factor each polynomial completely. $$ (a-b)^{3}-(a+b)^{3} $$

Galileo 's formula describing the motion of freely falling objects is $$ d=16 t^{2} $$ The distance d in feet an object falls depends on the time \(t\) elapsed, in seconds. (This is an example of an important mathematical concept, the function.) When you substituted 256 for \(d\) and solved the formula for \(t\) in Exercise \(79,\) you should have found two solutions: 4 and \(-4 .\) Why doesn't \(-4\) make sense as an answer?

Find each product. \(2 x^{2}\left(x^{2}+3 x+5\right)\)
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