/*! 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 88 Find the LCM. $$80,96,108$$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 88

Find the LCM. $$80,96,108$$

Short Answer

Expert verified
The LCM is 4320.

Step by step solution

01

- Prime Factorization

First, find the prime factorization of each number:- 80: \(80 = 2^4 \times 5\)- 96: \(96 = 2^5 \times 3\)- 108: \(108 = 2^2 \times 3^3\)
02

- Identify the Highest Power of Each Prime

Next, identify the highest power of each prime number that appears in the factorizations:- For 2: the highest power is \(2^5\)- For 3: the highest power is \(3^3\)- For 5: the highest power is \(5^1\)
03

- Multiply the Highest Powers

Now, multiply these highest powers together to find the LCM:\[ \text{LCM} = 2^5 \times 3^3 \times 5 = 32 \times 27 \times 5 = 4320 \]

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 a key step in finding the LCM. It involves breaking down each number into its prime factors. Primes are numbers greater than 1 that have no positive divisors other than 1 and themselves. To factorize, you keep dividing the number by the smallest possible prime until you finish.

In our example:
  • For 80, we divide by 2 to get: 80 becomes 24 脳 5
  • For 96, we divide by 2 and 3: 96 becomes 25 脳 3
  • For 108, we divide by 2 and 3: 108 becomes 22 脳 33
These factorizations will help us identify which primes and how many of each prime we need to compute the LCM.
Highest Power
The highest power is important because the LCM must include the maximum occurrence of each prime factor from all factorizations. Think of it as ensuring every common divisor is covered fully.

In our example, we see:
  • The highest power of 2 is 25, from 96
  • The highest power of 3 is 33, from 108
  • The highest power of 5 is 51, from 80
By taking the highest power of these primes, you ensure that the LCM is able to uniformly divide each of the original numbers.
Multiplication of Primes
The final step in finding the LCM is multiplying these highest powers together. This step consolidates all the primes at their highest powers to form the LCM.
  • From our example, we multiply:
    LCM = 25 脳 33 脳 51

Breaking it down, we calculate:
\[ \text{LCM} = 32 \times 27 \times 5 = 4320 \ \]
Each of these calculations multiplies one prime with its highest power and ensures all prime factors are represented in the final result. Thus, the least common multiple of 80, 96, and 108 is 4320.

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

African Artistry. In southern Africa, the design of every woven handbag, or gipatsi (plural, sipatsi) is created by repeating two or more geometric patterns. Each pattern encircles the bag, sharing the strands of fabric with any pattern above or below. The length, or period, of each pattern is the number of strands required to construct the pattern. For a gipatsi to be considered beautiful, each individual pattern must fit a whole number of times around the bag. (image cannot copy) A weaver is using two patterns to create a gipatsi. Pattern A is 10 strands long, and pattern \(\mathrm{B}\) is 3 strands long. What is the smallest number of strands that can be used to complete the gipatsi? (image cannot copy)

Boating. Shoreline Travel operates a 3 -hr paddleboat cruise on the Missouri River. If the speed of the boat in still water is \(12 \mathrm{mph}\), how far upriver can the pilot travel against a 5 -mph current before it is time to turn around?

Find equivalent expressions that have the LCD. $$\frac{7}{3 x^{4} y^{2}}, \frac{4}{9 x y^{3}}$$

Find the LCM. $$9 x^{3}-9 x^{2}-18 x, 6 x^{5}-24 x^{4}+24 x^{3}$$

The length of a rectangular picture window is 3 yd greater than the width. The area of the rectangle is \(10 \mathrm{yd}^{2} .\) Find the perimeter.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.