Problem 69 - Fundamentals of Mathematics

Question

(Section 3.6) Find the least common multiple of \(6,16,\) and 72 .

Step-by-Step Solution

Answer

The least common multiple of 6, 16, and 72 is 144.

Step 1: Prime Factorization

First, we find the prime factorization of each number. - The prime factorization of 6 is: \(6 = 2^1 \times 3^1\)- The prime factorization of 16 is: \(16 = 2^4\)- The prime factorization of 72 is: \(72 = 2^3 \times 3^2\)

Step 2: Identify Maximum Exponents

Next, we determine the highest power of each prime number present in any of the factorizations. - For the prime number 2, the highest power is \(2^4\) (from 16).- For the prime number 3, the highest power is \(3^2\) (from 72).

Step 3: Calculate the LCM

To find the least common multiple, we multiply the highest powers of all prime numbers identified in Step 2. The LCM is calculated as:\[ \text{LCM} = 2^4 \times 3^2 \]Calculating this gives:\[ \text{LCM} = 16 \times 9 = 144 \]

Key Concepts

Prime Factorization

Prime factorization is an essential method in finding the least common multiple (LCM) of a set of numbers. It breaks down a number into the product of its prime factors, which are numbers that have no divisors other than 1 and themselves. For instance, when we look at the number 6, we factor it into the smallest possible prime numbers: \( 6 = 2 \times 3 \). Here, both 2 and 3 are prime.
This method is crucial because it allows us to see the building blocks of each number. Each number in our example is broken down in this way:

• The prime factorization of 6 is \( 2^1 \times 3^1 \).
• The prime factorization of 16 is \( 2^4 \).
• The prime factorization of 72 is \( 2^3 \times 3^2 \).

These factorizations help us identify the essential components needed to calculate the LCM by comparing the exponents of the prime numbers involved.

Highest Power

After establishing the prime factors, the next step involves determining the highest power of each prime number. This ensures that the LCM contains at least as many of every prime factor as any of the numbers being considered.
In our set of numbers \( 6, 16, \) and \( 72 \), we look at the powers of each prime number:

• For the prime number 2, we have powers \( 2^1, 2^4, \) and \( 2^3 \). The highest power here is \( 2^4 \) from the number 16.
• For the prime number 3, we have powers \( 3^1 \) and \( 3^2 \). The highest power is \( 3^2 \) from the number 72.

By taking these highest powers, we ensure that the LCM can be evenly divided by each of our original numbers, encompassing all necessary prime factors in sufficient quantities.

Step by Step Solution

With this understanding in place, calculating the LCM becomes a matter of combining these highest powers into one expression. We multiply the highest powers we identified:To find the LCM of \( 6, 16, \) and \( 72 \), we multiply the highest powers:\[ \text{LCM} = 2^4 \times 3^2 \]
Calculate this step by step:

• First, calculate \( 2^4 \), which is 16.
• Then, calculate \( 3^2 \), which is 9.
• Finally, multiply these results: \( 16 \times 9 = 144 \).

The LCM is therefore 144, which is the smallest number that can be divided evenly by each of our original numbers. This thorough step-by-step method ensures accuracy and understanding of how the LCM is formulated.