/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Find the greatest common factor.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 3

Find the greatest common factor. $$4,16,32$$

Short Answer

Expert verified
The greatest common factor of 4, 16, and 32 is 4.

Step by step solution

01

Perform Prime Factorization

Carry out prime factorization of each of the numbers. The prime factorization of 4 is \(2 \times 2\), that of 16 is \(2 \times 2 \times 2 \times 2\), and 32 is \(2 \times 2 \times 2 \times 2 \times 2\).
02

Identify Common Prime Numbers

From the prime factors of 4, 16, and 32, identify the common prime numbers. Here, the common prime is 2. It appears twice in 4, four times in 16 and five times in 32.
03

Determine GCF

To calculate the GCF, one must take the lowest power of the common primes. Since 2 is the common prime and appears twice in 4, which is the least of all, the greatest common factor of 4, 16, and 32 is \(2 \times 2\) which equals 4.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Prime Factorization
Prime factorization is the process of breaking down a number into its smallest prime number components. It's much like finding the building blocks of a number. Each number can be expressed as a product of prime numbers, which are numbers greater than 1 that have no divisors other than 1 and themselves.
  • For example, the number 4 can be broken down into: \(2 \times 2\).
  • Similarly, the number 16 can be decomposed into \(2 \times 2 \times 2 \times 2\).
  • Finally, 32 can be expressed as \(2 \times 2 \times 2 \times 2 \times 2\).

By representing numbers in this way, it becomes easier to analyze and find common factors by comparing their prime components.
Common Prime Numbers
Once each number is broken down into its prime factors, the next step is to identify the common prime numbers among them. This step involves checking which prime numbers appear in the prime factorization of all the numbers you鈥檙e examining.

In our example, the numbers 4, 16, and 32 all include the prime number 2.
  • In 4, the number 2 appears twice.
  • In 16, 2 shows up four times.
  • In 32, it appears five times.

Thus, the common prime number in this scenario is 2. Recognizing these common numbers is crucial for calculating the greatest common factor.
Least Power of Common Primes
Having identified the common prime numbers, the final step in finding the greatest common factor is to choose the least power of these common primes. The greatest common factor is constructed by using the smallest power, as this power appears in each number's prime factorization.

  • In 4, the power of 2 is \(2^2\).
  • In 16, the power of 2 increases to \(2^4\).
  • In 32, it's further raised to \(2^5\).

To determine the GCF, take the smallest instance: \(2^2\), which corresponds to 4. Therefore, the greatest common factor of 4, 16, and 32 is indeed 4. Utilizing this approach ensures that the GCF is accurately derived from the least power common to all terms.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$c^{2}-a^{2}+8 a-16$$

Factor. $$-26 x+6 x^{2}-20$$

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$a^{2}+6 a+9-b^{2}$$

Factor. If the polynomial is prime, so indicate. $$-12 t^{2}+1+4 t$$

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$u^{2}-3 v^{2}+2 u v$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.