Problem 20 - Prealgebra

Question

Find the least common multiple (LCM) of each pair of numbers or monomials. $$14,28$$

Step-by-Step Solution

Answer

The LCM of 14 and 28 is 28.

Step 1: Identify Prime Factors of Each Number

First, we need to find the prime factorization of each number. \(14\) can be broken down into prime factors as \(14 = 2 \times 7\). Similarly, \(28\) can be factorized into \(28 = 2^2 \times 7\).

Step 2: Compare Prime Factors

Next, we compare the prime factors of both numbers to determine the maximum power of each prime number that appears in either number's factorization. For \(2\), the highest power is \(2^2\) (from \(28\)), and for \(7\), it is \(7^1\) (both have \(7^1\)).

Step 3: Calculate the Least Common Multiple

The LCM is found by multiplying the highest powers of all prime factors identified. Multiply these together: \(2^2 \times 7^1 = 4 \times 7 = 28\). The least common multiple of \(14\) and \(28\) is therefore \(28\).

Key Concepts

Prime Factorization

Prime factorization is an essential concept in mathematics where any integer greater than 1 is expressed as a product of prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
For instance, consider the number 14. To find its prime factorization, we look for the smallest prime that can divide it, which is 2. Hence, we write:

• 14 divided by 2 equals 7.
• The next factorization is 7, which is already a prime number.

Therefore, the prime factorization of 14 is expressed as \(14 = 2 \times 7\).
Similarly, for 28, which can be broken down as:

• 28 divided by 2 equals 14, which further breaks down to \(14 = 2 \times 7\).
• Thus, 28 equals \(2^2 \times 7\).

Recognizing the prime factors helps in many areas of number theory and in finding the least common multiple.

Number Theory

Number theory delves into properties and relationships of numbers, particularly integers. It helps in tasks like finding the Least Common Multiple (LCM), which is the smallest number divisible by each of a given set of numbers.
To find the LCM using prime factorization:

• Identify all prime factors of the numbers involved.
• Take the highest power of each prime number that appears in the factorization of either number.

For example, with numbers 14 and 28, we identify prime factors \(2\) and \(7\). Compare their powers:

• The highest power of 2 is \(2^2\) (coming from 28).
• The highest power of 7 is \(7^1\) (shared by both numbers).

The LCM is then calculated as \(2^2 \times 7 = 28\). This result means 28 is the smallest number that all the given numbers (14 and 28) can divide without a remainder.

Mathematics Education

Mathematics education emphasizes the importance of understanding concepts like prime factorization and LCM for developing problem-solving skills. These concepts are used across a range of real-world applications, from simplifying fractions to calculating common periods in time management.
It's important for students to grasp these basics because they build upon knowledge incrementally. For example, understanding prime factorization now helps students solve more complex problems, like finding the LCM or GCD in algebra, later on.
Educators often use visual aids and interactive activities to teach these concepts, ensuring students understand the logic and process behind prime factorization and LCM without just memorizing steps. By engaging with number theory early on, students develop a strong foundation in mathematical reasoning, preparing them for more advanced studies and everyday math problem-solving.