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Chapter 3: Problem 12

Find the LCM of each set of numbers. $$ 21,27 $$

Short Answer

Expert verified
The LCM of 21 and 27 is 189.

Step by step solution

01

Prime Factorization of Each Number

First, find the prime factorization of each number.- For 21: 21 = 3 脳 7- For 27: 27 = 3^3
02

Identify the Highest Power of Each Prime

List all the prime numbers that appear in the factorizations, and choose the highest power of each.- Prime number 3: The highest power is 3^3.- Prime number 7: The highest power is 7.
03

Calculate the LCM

Multiply the highest powers of all prime numbers from step 2.LCM = 3^3 脳 7 = 27 脳 7 = 189

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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 the process of breaking down a composite number into its prime factors. A prime number is a number greater than 1 that only has two divisors: 1 and itself. For example, 2, 3, 5, 7, and 11 are all prime numbers. To break a number like 21 into its prime factors, we find that 21 = 3 脳 7. Here, 3 and 7 are both primes. Similarly, the number 27 can be broken down into prime factors as 27 = 3^3. This means 3 is used three times in multiplication.
Highest Power
When finding the Least Common Multiple (LCM), we need to determine the highest powers of the prime factors that appear in the numbers. Each prime appears in different powers in the factorizations. For instance, in the prime factorization of 21 (3 脳 7), the prime 3 appears as 3^1 and 7 appears as 7^1. For 27 (3^3), the prime 3 appears as 3^3. To find the LCM, we take the highest power of each prime number involved. The highest power of 3 is 3^3, and for 7 it is 7^1.
Multiplication
To find the LCM, once we have identified the highest powers of the prime numbers, we simply multiply them. In our example, the highest powers are 3^3 and 7^1. Thus, we calculate the LCM by performing the multiplication: \( 3^3 \times 7^1 = 27 \times 7 = 189 \). So, the LCM of 21 and 27 is 189. This product gives the smallest multiple that is common to both original numbers.

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