Problem 59 - College Algebra with Modeling and Visualization

Question

Find the least common multiple. $$12,18$$

Step-by-Step Solution

Answer

The least common multiple of 12 and 18 is 36.

Step 1: List the Factors

To find the least common multiple (LCM) of 12 and 18, start by listing the prime factors of each number. The prime factors of 12 are 2, 2, and 3, while the prime factors of 18 are 2, 3, and 3.

Step 2: Identify the Highest Powers

Next, identify the highest power of each prime number that appears in the factorizations. For the number 2, the highest power available in the numbers 12 and 18 is 2^2 from 12. For the number 3, the highest power available is 3^2 from 18.

Step 3: Calculate the Least Common Multiple

To find the LCM, multiply the highest powers of all prime numbers together. LCM = 2^2 * 3^2 = 4 * 9 = 36.

Key Concepts

Understanding Prime Factorization

Prime factorization is breaking down a number into its simplest building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. To determine the least common multiple (LCM) of numbers, begin by identifying the prime factors of each number involved. For instance, let's take the numbers 12 and 18.

• The prime factors of 12: Since 12 breaks down to 2 × 2 × 3, its prime factors are 2 and 3.
• The prime factors of 18: 18 breaks down to 2 × 3 × 3, so its prime factors are 2 and 3.

Remember that each number can be expressed as a product of these prime factors, which makes it easier to find the LCM, as different numbers often share some of the same prime factors. This shared property allows for a streamlined calculation when establishing relationships like the LCM.

Recognizing Highest Powers

Once you have identified the prime factors of each number, the next step is recognizing the highest power of each prime number within those factorizations. This involves comparing each prime that appears in the factors of the given numbers and noting the maximum times each prime appears. Use this to find the LCM.

• Observe the prime 2: In the numbers 12 and 18, the highest power in which 2 appears is 2² from 12.
• Look at the prime 3: Here, the highest power for 3 is 3² from 18.

To construct the LCM, each prime number from the factorization considers only the highest power found among the numbers. This method ensures that the LCM covers every multiple of each factorized number without redundancy.

Applying Multiplication for the LCM

Now that we have the highest powers of the prime factors from our factorizations, we move on to find the least common multiple using multiplication. The LCM is calculated by multiplying these highest powers together. This simple step yields a number that can be divided by each of the original numbers with no remainder.

• Using the primes we've examined, we perform the multiplication: LCM = 2² × 3².
• Solving this gives us: 2² = 4 and 3² = 9, thus 4 × 9 = 36.

The result, 36, is the smallest number that both 12 and 18 divide into without leaving a remainder. By ensuring each prime factor is represented at its highest power, the LCM can define a shared multiple that encompasses the full breadth of the factors involved.