Problem 24 - Elementary Algebra

Question

For the following problems, find the least common multiple of given numbers. 6, 9, 12, 18

Step-by-Step Solution

Answer

Answer: The least common multiple of the numbers 6, 9, 12, and 18 is 36.

Step 1: Prime factorization of each number

First, we need to find the prime factorization of each number. The prime factorization is representing a number as the product of its prime factors. 6 = 2*3 9 = 3^2 12 = 2^2*3 18 = 2*3^2

Step 2: Identify the highest power of each prime factor

Next, we should identify the highest power of each prime factor that appears in the factorization of these numbers. For prime factor 2, the highest power is 2^2 (from number 12). For prime factor 3, the highest power is 3^2 (from numbers 9 and 18).

Step 3: Calculate the LCM

Finally, we can calculate the least common multiple (LCM) by multiplying the highest power of each prime factor together. This gives us the smallest positive integer that each of the numbers divides. LCM = 2^2 * 3^2 = 4 * 9 = 36. The least common multiple of 6, 9, 12, and 18 is 36.

Key Concepts

Prime Factorization

When we talk about prime factorization, we are breaking down a number into its most basic building blocks: prime numbers.It's similar to identifying the DNA of a number. Prime numbers are those that have no divisors other than 1 and themselves.
For instance, the prime factorization of 6 is 2 and 3, giving us the product 2 \( \times \) 3. This process is applied to each number we're examining.

• 6 can be factored into 2 \( \times \) 3.
• 9 is factored into 3^2 (since 9 is 3 \( \times \) 3).
• 12 is factored into 2^2 \( \times \) 3.
• 18 breaks down into 2 \( \times \) 3^2.

Prime factorization helps us see the numbers in a form that makes it easier to understand their properties and relationships, especially when finding the least common multiple.

Highest Power of Prime Factors

Once the numbers are expressed in terms of their prime factors, the next step involves identifying the highest power of each prime factor across all numbers. Imagine you are collecting the strongest form of each prime number that appears.
This is crucial because the least common multiple demands that we account for every prime factor at its maximum contribution.

• For the prime factor 2, the highest power is 2^2, derived from 12.
• For the prime factor 3, the highest power is 3^2, which we see in both 9 and 18.

This step ensures that when we multiply these highest powers, each number in our set can divide the resulting product without leaving a remainder.

Multiplication of Prime Factors

Here we bring it all together by multiplying the highest powers of the prime factors we identified.This step determines the least common multiple—the smallest number that each of our original numbers can divide evenly.
Essentially, it’s as if we’re constructing the least common multiple from these prime factor blocks.

• First, multiply 2^2, which gives 4.
• Then, multiply 3^2, yielding 9.
• Finally, multiply these products: 4 \( \times \) 9 = 36.

The result, 36, is the least common multiple for 6, 9, 12, and 18. This number, 36, can be divided by each of the original numbers without any remainder, perfectly demonstrating the beauty of prime factorization in solving real-world problems.