Find The Number Between 345 And 380 Divisible By 3 With A Digit Sum Of 18

Embark on a captivating journey into the realm of numbers as we unravel a fascinating mathematical puzzle. Our quest centers around identifying a specific number nestled between the boundaries of 345 and 380. However, this is no ordinary numerical pursuit. This elusive number possesses a unique set of characteristics that elevate it beyond the mundane and into the realm of mathematical intrigue. To successfully unmask this enigmatic figure, we must delve into the depths of divisibility rules and the intriguing interplay of numerical digits.

The Divisibility Rule of 3: A Key to Unlocking the Mystery

Our primary clue lies in the fact that the number we seek is divisible by 3. This seemingly simple statement unlocks a powerful mathematical principle – the divisibility rule of 3. This rule serves as a beacon, guiding us through the numerical landscape and illuminating potential candidates that fit our criteria. The divisibility rule of 3 states that a number is divisible by 3 if the sum of its digits is also divisible by 3. This elegant rule transforms a seemingly complex problem into a manageable task. Instead of performing lengthy division calculations, we can simply add the digits of a number and check if the sum is divisible by 3. This clever shortcut significantly streamlines our search, allowing us to efficiently sift through potential candidates and narrow down our options.

Let's illustrate this principle with a few examples. Consider the number 351. The sum of its digits (3 + 5 + 1) is 9, which is indeed divisible by 3. Therefore, according to the divisibility rule, 351 is also divisible by 3. Conversely, let's examine the number 352. The sum of its digits (3 + 5 + 2) is 10, which is not divisible by 3. Consequently, 352 is not divisible by 3. This simple yet powerful rule empowers us to quickly identify multiples of 3, significantly accelerating our quest to find the mystery number.

In our pursuit, we will systematically examine the numbers between 345 and 380, applying the divisibility rule of 3 as our primary filter. Any number whose digits do not sum to a multiple of 3 can be immediately discarded, saving us valuable time and effort. This strategic approach ensures that we focus our attention on the most promising candidates, increasing our chances of successfully solving the puzzle.

The Sum of Digits: A Further Refinement

Adding another layer of complexity to our challenge, we are informed that the sum of the digits of our mystery number must equal 18. This constraint acts as a powerful sieve, further refining our search and eliminating numbers that, while divisible by 3, do not meet this specific criterion. The requirement that the digits sum to 18 significantly narrows down the pool of potential candidates, bringing us closer to pinpointing the elusive number.

This constraint compels us to analyze the composition of each potential candidate, scrutinizing the individual digits and their collective sum. We are not merely looking for a number divisible by 3; we are seeking a number whose digits, when combined, yield the precise sum of 18. This added layer of complexity transforms our puzzle from a simple divisibility test into a fascinating exercise in numerical analysis.

Consider the number 369. It is divisible by 3, but the sum of its digits (3 + 6 + 9) is 18. Therefore, 369 satisfies our specific digit sum criterion. On the other hand, consider the number 357. It is also divisible by 3, but the sum of its digits (3 + 5 + 7) is 15, falling short of our target sum of 18. Consequently, 357 can be eliminated from our list of potential candidates. This meticulous process of digit summation allows us to progressively narrow down our options, honing in on the number that perfectly matches all the given criteria.

By combining the divisibility rule of 3 with the digit sum requirement, we create a powerful two-pronged approach to solving our puzzle. The divisibility rule acts as an initial filter, weeding out numbers that are not multiples of 3. The digit sum criterion then serves as a finer sieve, selecting only those numbers whose digits add up to the magic number 18. This synergistic approach ensures that we efficiently and accurately identify the mystery number, transforming a potentially daunting task into a manageable and enjoyable challenge.

The Quest for the Number: Applying the Clues

Equipped with our understanding of the divisibility rule of 3 and the digit sum requirement, we are now ready to embark on the final stage of our quest. We will systematically examine the numbers between 345 and 380, applying our two key criteria to identify the elusive mystery number. This methodical approach will ensure that we leave no stone unturned in our pursuit of the solution.

Let's begin by considering the numbers in the 340s. We can quickly eliminate numbers like 346 and 347, as they are not divisible by 3. The number 348, however, is divisible by 3. But the sum of its digits (3 + 4 + 8) is 15, which does not equal 18. So, 348 can also be discarded. Continuing in this manner, we reach the number 351. It is divisible by 3, but the sum of its digits (3 + 5 + 1) is only 9, far short of our target. As we progress, we encounter 354, divisible by 3, with a digit sum of 12 – still not 18.

Moving into the 360s, we find the number 360. While divisible by 3, the digit sum is a mere 9. The number 363 follows, again divisible by 3, but the digit sum is 12. As we continue our exploration, we arrive at the number 369. This is a pivotal moment in our quest. Not only is 369 divisible by 3, but the sum of its digits (3 + 6 + 9) is exactly 18! Could this be our mystery number? It certainly satisfies both of our key criteria. However, we must remain vigilant and continue our search to ensure that we have identified the only number that fits the description.

Venturing into the 370s, we encounter numbers like 372 and 375, both divisible by 3. However, their digit sums (12 and 15, respectively) fall short of 18. As we approach the upper limit of our range, we find that no other number between 375 and 380 satisfies both the divisibility rule of 3 and the digit sum requirement. This confirms our earlier suspicion: 369 is indeed the unique number that perfectly fits all the given criteria.

The Grand Reveal: The Mystery Number Unmasked

After a meticulous and systematic search, we have successfully unmasked the mystery number. Through the application of the divisibility rule of 3 and the digit sum requirement, we have navigated the numerical landscape and pinpointed the elusive figure. The number that lies between 345 and 380, divisible by 3, and possessing a digit sum of 18, is none other than 369. This triumphant conclusion marks the end of our mathematical quest, a journey that has taken us through the intriguing world of divisibility rules and digit analysis.

The unveiling of the number 369 as the solution to our puzzle highlights the power of mathematical principles in solving seemingly complex problems. By understanding and applying fundamental rules, we can unravel numerical mysteries and gain a deeper appreciation for the elegance and precision of mathematics. This exercise serves as a reminder that mathematics is not merely a collection of formulas and equations; it is a powerful tool for problem-solving and critical thinking, applicable to a wide range of real-world scenarios.

The successful resolution of this puzzle also underscores the importance of methodical and systematic approaches to problem-solving. By breaking down the problem into smaller, manageable steps, and by applying logical reasoning and deduction, we were able to efficiently narrow down the possibilities and arrive at the correct solution. This approach is not only valuable in mathematics but also in various aspects of life, where complex challenges often require a structured and analytical approach.

In conclusion, our journey to uncover the mystery number between 345 and 380 has been a rewarding experience. We have not only identified the number 369 as the solution but also reinforced our understanding of fundamental mathematical principles and problem-solving strategies. This exercise serves as a testament to the beauty and power of mathematics, a discipline that continues to challenge and inspire us.