Determining the Fourth Number using GCD and LCM

Introduction

In mathematics, the greatest common divisor (GCD) and the least common multiple (LCM) are fundamental concepts that play a crucial role in number theory. This article will explore how to determine a fourth number when given the GCD, LCM, and some known numbers. We will solve a specific problem where the GCD of four numbers is 8 and the LCM is 1680, with three of the numbers being 16, 24, and 56. Let's break down the process step-by-step.

Step-by-Step Solution

First, we need to calculate the product of the three known numbers (16, 24, and 56).

Step 1: Calculate the Product of the Known Numbers

To find the product of the known numbers, we perform the multiplication: 16 * 24 * 56 Breaking it down step by step: 1. 16 * 24 384 2. 384 * 56 21504 So, the product of 16, 24, and 56 is 21504.

Step 2: Use the Relationship between GCD, LCM, and the Product of the Numbers

The relationship between the GCD, LCM, and the numbers is given by the formula: text{GCD}(a, b, c, d) * text{LCM}(a, b, c, d) a * b * c * d Let the fourth number be (d). Given that the GCD of the four numbers is 8 and the LCM is 1680, we have: 8 * 1680 16 * 24 * 56 * d Calculating the left side: 8 * 1680 13440

Step 3: Set Up the Equation

Now we can set up the equation based on the relationship: 13440 21504 * d

Step 4: Solve for (d)

To find (d), we can rearrange the equation and solve for (d): d frac{13440}{21504} Calculating the fraction: d frac{13440 div 6720}{21504 div 6720} frac{2}{3.2} frac{20}{32} frac{5}{8} Since we know the GCD is 8, we multiply (d) by the GCD to find the actual fourth number: d frac{5}{8} * 8 5

Final Answer

Thus, the fourth number is 5.

Conclusion

We followed a systematic approach to determine the fourth number using the GCD and LCM of the numbers. The key steps involved calculating the product of the known numbers, using the GCD-LCM relationship, setting up the equation, and solving for the unknown number. This method can be applied to similar problems to find unknown numbers when the GCD and LCM are given.

Note: Upon re-evaluating the solution, the fourth number should actually be 40, not 5. The detailed calculation is as follows:

Correct Calculation

Revisiting the LCM and GCD relationships, the correct approach involves ensuring the multiplication is accurate and re-calculating the actual fourth number. By re-evaluating the LCM and GCD relationships, we confirm that the fourth number is indeed 40, leading to the correct LCM and GCD values.

Key Factors and Relevant Numbers

The problem involves the numbers (16), (24), and (56). Let's factorize these numbers and identify their greatest common divisor (GCD) and least common multiple (LCM): 16 1, 2, 4, 8, 16 24 1, 2, 3, 4, 6, 8, 12, 24 56 1, 2, 4, 7, 8, 14, 28, 56 40 1, 2, 4, 5, 8, 10, 20, 40 The GCD of 16, 24, and 56 is 8, and the LCM of these numbers is 168. To find the fourth number, we ensure the LCM and GCD values hold true.

Contrast with Initial Solution

By dividing 16, 24, and 56 by their GCD (8), we get the numbers 2, 3, and 7. Multiplying these by the GCD (8) and then by 5 (as (1680 div 336 5)), we confirm the fourth number is 40.

Conclusion and Final Answer

The fourth number is thus 40, correcting the initial solution.

Related Problems and Concepts

Understanding GCD and LCM can be applied to various problems in mathematics. Familiarity with these concepts is essential for solving number theory and algebraic problems.