Problem 12 - Fundamentals of Mathematics

Question

Find the LCM of the following numbers. 14 and 28

Step-by-Step Solution

Answer

The LCM of 14 and 28 is 28.

Step 1: Prime Factorization

First, we need to find the prime factorization of each number. For 14, the prime factors are 2 and 7, as 14 can be expressed as \(14 = 2 \times 7\). For 28, the prime factors are also 2 and 7, as 28 can be expressed as \(28 = 2^2 \times 7\).

Step 2: Identify the Highest Powers

To find the least common multiple (LCM), we need to take the highest power of each prime number found in the factorizations. From 14 and 28, we have the primes 2 and 7. The highest power of 2 is \(2^2\) (from 28) and the highest power of 7 is \(7^1\) (both 14 and 28).

Step 3: Calculate the LCM

The LCM is found by multiplying the highest powers of all prime factors together. Using the highest powers from step 2, we calculate: \[ LCM = 2^2 \times 7^1 = 4 \times 7 = 28.\]

Step 4: Verify the LCM

To ensure correctness, verify that 28 is divisible by both original numbers. Dividing 28 by 14 gives 2, and dividing 28 by 28 gives 1, confirming 28 is indeed the smallest number divisible by both.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. This is like finding the unique way to express the number as a product of prime numbers. A prime number is a number greater than 1 that has no other factors besides 1 and itself. For instance, consider the number 14. It can be factored into 2 and 7, both of which are prime numbers, as shown by the expression:

• 14 = 2 x 7

Now, take the number 28. It can also be factored using the same prime numbers but note that the factor of 2 appears twice:

• 28 = 2 x 2 x 7 = 2^2 x 7

This differentiates 28's prime factorization from that of 14 due to the repetition of 2. Understanding this concept is crucial because it helps in finding the least common multiple of numbers.
Prime factorization sets the stage for evaluating multiple numbers in terms of their prime roots, simplifying processes like finding the least common multiple (LCM).

Powers of Prime Numbers

When finding the least common multiple (LCM), powers of prime numbers play an essential role. Let’s take a closer look at this. Powers of a number refer to the number of times a prime factor appears in the factorization. For example:

• In 28’s factorization, 2 appears twice (\(2^2\)).
• In both 14 and 28, 7 appears once (\(7^1\)).

When calculating the LCM, identifying the highest power of each prime factor from all numbers involved is fundamental. In this exercise:

• The highest power of 2 is \(2^2\)
• The highest power of 7 is \(7^1\)

These powers are then multiplied to find the LCM.
Using powers allows us to neatly combine factorizations, capturing their essence without missing any number that might divide each of the numbers. This systematic approach ensures no prime is overlooked, especially when multiple numbers are involved.

Math Verification

After calculating a mathematical result, it's important to verify the accuracy of that result. Math verification serves as a check to ensure calculations are correct. Once we've determined the least common multiple (LCM) in this exercise, the next step is to verify:

• We calculate \(LCM = 2^2 \times 7^1 = 4 \times 7 = 28\)

Verification involves checking that the calculated LCM is divisible evenly by the original numbers. Thus, we need to ensure that both 14 and 28 divide into 28 without leaving a remainder:

• 28 ÷ 14 = 2
• 28 ÷ 28 = 1

The results above confirm that 28 is the smallest number divisible by both, certifying our LCM calculation as correct.
Additionally, math verification fosters confidence in calculations, assuring that the number found truly fulfills the required mathematical properties. In fields like mathematics and engineering, these checks are invaluable for maintaining accuracy.