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Chapter 3: Problem 26

Find the LCM of each set of numbers. $$ 8,16,22 $$

Short Answer

Expert verified
The LCM of 8, 16, and 22 is 176.

Step by step solution

01

List the Prime Factors

First, find the prime factors of each number. For 8: $$ 8 = 2^3 $$For 16: $$ 16 = 2^4 $$For 22: $$ 22 = 2 \times 11 $$
02

Identify Highest Powers

Next, identify the highest powers of all prime numbers present in the factorizations.Prime factors: 2, 11Highest powers: $$ 2^4 $$ (from 16) $$ 11^1 $$ (from 22)
03

Calculate the LCM

Multiply the highest powers of all the prime numbers.LCM: $$ 2^4 \times 11^1 = 16 \times 11 = 176 $$

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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 expressing a number as the product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For instance, let's break down the numbers from our exercise:
  • For 8: Since 8 can be divided by 2 without leaving a remainder, we repeatedly divide it by 2 to get: 8 = 2 脳 2 脳 2, or in exponent form, 8 = 2^3.
  • For 16: Similar to 8, we continuously divide by 2: 16 = 2 脳 2 脳 2 脳 2, or 16 = 2^4.
  • For 22: 22 is divided by 2 once and leaves 11, a prime number, so 22 = 2 脳 11.
Understanding prime factorization is essential because it lays the groundwork for finding the LCM and performing many other mathematical operations.
Highest Powers
Once you have the prime factorizations, the next step is to identify the highest powers of each prime number involved. This helps in ensuring that the least common multiple (LCM) captures all factors needed for the given numbers.

For the numbers 8, 16, and 22, note the prime factors: 2 and 11. Next, observe the exponents of these factors across all numbers:
  • The prime factor 2 has exponents 3 (in 8), and 4 (in 16).
  • The prime factor 11 has an exponent of 1 (only in 22).
We choose the highest exponent for each of those primes:
  • The highest power of 2 is 4 (from 16).
  • The highest power of 11 is 1 (from 22).
This step ensures that all prime factors are considered at their greatest influence when calculating the LCM.
Multiplication of Prime Factors
The final step in finding the LCM is to multiply the highest powers of all involved prime factors. This helps convene the influence of every number's prime factors and ensures no repeated counting of common factors.

Let's multiply the highest powers we identified:
  • From prime factor 2: The highest power is 4, i.e., 2^4 = 16.
  • From prime factor 11: The highest power is 1, i.e., 11^1 = 11.
Combining these, the calculation goes as follows:
  • LCM = 2^4 脳 11^1 = 16 脳 11 = 176.
Thus, 176 is the least common multiple of the numbers 8, 16, and 22, encapsulating the highest influences of each prime factor.

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