Problem 3 - Prealgebra

Question

Find the least common multiple (LCM) of each pair of numbers. $$10,14$$

Step-by-Step Solution

Answer

The LCM of 10 and 14 is 70.

Step 1: List the Prime Factors

First, determine the prime factors of each number.\( 10 \) breaks down into \( 2 \times 5 \), and \( 14 \) breaks down into \( 2 \times 7 \).

Step 2: Identify the Greatest Power of Each Prime

List all prime numbers found in both factorizations and determine the highest power of each.\( 2 \) is raised to the first power in both factors, \( 5 \) is a factor of \( 10 \) (raised to the first power), and \( 7 \) is a factor of \( 14 \) (raised to the first power).

Step 3: Multiply Highest Powers of Each Prime

To find the LCM, multiply the highest powers of each prime number: \( 2^1 \times 5^1 \times 7^1 = 70 \). This product is the least common multiple.

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. It involves breaking down a composite number into the smallest building blocks that multiply together to create the original number. For example, in the case of 10 and 14, these numbers are decomposed as follows:

• 10 is equal to \( 2 \times 5 \), where both 2 and 5 are primes.
• 14 is equal to \( 2 \times 7 \), where both 2 and 7 are primes.

Prime factorization is like finding the recipe for a number using the essential, undividable ingredients - the prime numbers. By identifying these prime factors, we create a solid foundation for methods like finding the Least Common Multiple (LCM), which we'll need to solve various mathematical problems like the one presented here.

Greatest Power of Prime

In finding the Least Common Multiple (LCM), it is important to identify the greatest power of each prime number involved in the factorization of the given numbers. This step ensures every necessary ingredient is included in the final multiplication to guarantee that both original numbers can divide the LCM without leaving a remainder.Let's look at how we did this in our example:

• The prime factors of 10 are 2 and 5, both raised to the power of 1 (or \( 2^1 \) and \( 5^1 \)).
• The prime factors of 14 are 2 and 7, both similarly raised to the power of 1 (or \( 2^1 \) and \( 7^1 \)).

When choosing the greatest power, it means taking the 2, 5, and 7 at their existing highest powers from among the factorizations. This ensures that the product we create is the smallest possible number that each original number can fully divide, which is crucial to finding the LCM.

Multiplication of Prime Numbers

The final step in determining the Least Common Multiple requires multiplying the greatest power of each prime number identified in the previous steps. This multiplication brings together all the elements needed to form the smallest common product, accessible to all integers involved.In our example, 10 and 14 are broken down into:

• \(2^1\) (from both factorizations),
• \(5^1\) (from 10),
• \(7^1\) (from 14).

By multiplying these together, we perform the following calculation: \[ 2^1 \times 5^1 \times 7^1 = 70 \]Thus, 70 serves as the Least Common Multiple of 10 and 14, encompassing all necessary prime factors without excess. This foundational skill of multiplying prime powers is not only crucial for solving LCM problems but also has broader applications in arithmetic and algebra.