Problem 20 Find the LCM. 27 and 45... [FREE SOLUTION]

91影视

Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 4: Problem 20

Find the LCM. 27 and 45

Short Answer

Expert verified

The LCM of 27 and 45 is 135.

Step by step solution

- Prime Factorization

Determine the prime factorization of each number. For 27: \[ 27 = 3^3 \]For 45: \[ 45 = 3^2 \times 5 \]

- Identify Highest Powers

Identify the highest power of each prime factor from the factorizations. The highest powers are: \[ 3^3 \text{ and } 5^1 \]

- Calculate the LCM

Multiply the highest powers of the prime factors to find the LCM: \[ LCM = 3^3 \times 5^1 = 27 \times 5 = 135 \]

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 method used to break down a number into its basic building blocks鈥攑rime numbers. Prime numbers are those numbers that are only divisible by 1 and themselves, such as 2, 3, 5, and 7.
For instance, in the problem, we have two numbers: 27 and 45. To find the Least Common Multiple (LCM), we first break these numbers into their prime factors.

For 27, the prime factorization would be:
\[ 27 = 3 \times 3 \times 3 \text{ or } 3^3 \]
For 45, the prime factorization would be:
\[ 45 = 3 \times 3 \times 5 \text{ or } 3^2 \times 5 \]
Breaking numbers down to their primes is crucial because it allows us to see what basic 'ingredients' make up each number.

LCM Calculation

Once the numbers are broken down into their prime factors, we can use these factors to calculate the LCM. LCM is used to find the smallest multiple that is common to two or more numbers.
After determining the prime factors, we identify the highest power of each prime number that appears in the factorizations of the numbers involved.
In our exercise, for the numbers 27 and 45, the prime factors are:

For 27: \[ 3^3 \]
For 45: \[ 3^2 \times 5 \]

The highest power of each prime that appears in these factorizations is:

\[ 3^3 \] because 3 appears with the highest exponent (3)
\[ 5^1 \] because 5 appears with the highest exponent (1)

So the LCM is found by multiplying these highest powers together.

Highest Powers

Identifying the highest powers of each prime factor is an essential step in calculating the LCM. We look at the prime factorizations and note the highest exponent for each prime number.
In our given problem, the highest powers from the factorizations are:

\[ 3^3 \]
\[ 5^1 \]

We then multiply these highest powers to find the LCM:

\[ LCM = 3^3 \times 5^1 \]

In simpler terms, this gives us:
\[ LCM = 27 \times 5 = 135 \]
Breaking down and understanding this step is important, as it shows us how to combine prime factors utilizing their highest occurrences to ensure the result is the smallest common multiple.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the LCM. 27 and 45

Short Answer

Step by step solution

- Prime Factorization

- Identify Highest Powers

- Calculate the LCM

Key Concepts

Prime Factorization

LCM Calculation

Highest Powers

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Calculus

Theoretical and Mathematical Physics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.