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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

01

- Prime Factorization

Determine the prime factorization of each number. For 27: \[ 27 = 3^3 \]For 45: \[ 45 = 3^2 \times 5 \]
02

- Identify Highest Powers

Identify the highest power of each prime factor from the factorizations. The highest powers are: \[ 3^3 \text{ and } 5^1 \]
03

- Calculate the LCM

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

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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.

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