Three-Digit Numbers Divisible By 25 With Digit Sum Of 16
In the realm of mathematics, number theory presents a captivating array of problems that challenge our understanding of numerical relationships and properties. Among these problems, those involving divisibility and digit sums hold a special intrigue. This article delves into a specific problem of this nature: determining the count of three-digit numbers that satisfy two distinct criteria – having a digit sum of 16 and being divisible by 25. This exploration will not only provide a solution to the problem but also illuminate the underlying mathematical principles and techniques involved. Understanding the divisibility rules and digit sum properties is crucial in solving such problems, which often appear in mathematical competitions and aptitude tests. By carefully analyzing the conditions and applying logical reasoning, we can systematically narrow down the possibilities and arrive at the correct answer. This article aims to break down the problem into manageable steps, making the solution process clear and understandable for readers with varying levels of mathematical expertise.
The core question we aim to address is: How many three-digit numbers exist such that the sum of their digits equals 16, and the number itself is divisible by 25? This problem combines the concepts of digit sums and divisibility, requiring us to consider both aspects simultaneously. To solve this, we need to understand the conditions under which a number is divisible by 25 and how to find combinations of digits that add up to 16. The constraints of the problem, such as the number being a three-digit number, further limit the possibilities and guide our approach. Understanding the problem statement is the first step towards finding a solution. We must clearly define what we are looking for and what conditions must be met. This involves recognizing that we are dealing with a finite set of numbers (three-digit numbers) and that we have two specific criteria to satisfy: divisibility by 25 and a digit sum of 16. By carefully considering these elements, we can begin to formulate a strategy for solving the problem.
To effectively tackle this problem, we must first understand the divisibility rule for 25. A number is divisible by 25 if and only if its last two digits form a number that is divisible by 25. This means the last two digits can be 00, 25, 50, or 75. This rule significantly narrows down the possibilities for the three-digit numbers we are looking for. Instead of considering all three-digit numbers, we only need to focus on those ending in 00, 25, 50, or 75. This simplification is crucial in making the problem more manageable. Understanding the divisibility rule not only helps us identify potential candidates but also eliminates a large number of numbers that do not meet the criteria. This targeted approach is a key strategy in problem-solving, especially in mathematics. By applying the divisibility rule of 25, we can efficiently filter out numbers and concentrate on those that have a higher chance of satisfying the digit sum condition as well. This step is essential in streamlining our search for the solution.
Now that we have narrowed down the possibilities based on the divisibility rule of 25, we need to apply the digit sum condition. We are looking for three-digit numbers where the sum of the digits is 16. Let's consider each possible ending (00, 25, 50, and 75) separately:
- Numbers ending in 00: If the last two digits are 00, the first digit must be 16 to satisfy the digit sum condition. However, since the first digit must be a single digit (1 to 9), this case is not possible.
- Numbers ending in 25: If the last two digits are 25, their sum is 7. To reach a total digit sum of 16, the first digit must be 16 - 7 = 9. So, the number 925 is a potential candidate.
- Numbers ending in 50: If the last two digits are 50, their sum is 5. To reach a total digit sum of 16, the first digit must be 16 - 5 = 11. Again, since the first digit must be a single digit, this case is not possible.
- Numbers ending in 75: If the last two digits are 75, their sum is 12. To reach a total digit sum of 16, the first digit must be 16 - 12 = 4. So, the number 475 is another potential candidate.
By systematically analyzing each case, we have identified two numbers, 925 and 475, that satisfy both the divisibility rule of 25 and the digit sum condition. This methodical approach ensures that we do not miss any potential solutions and that we consider all possibilities within the given constraints. The digit sum condition acts as a further filter, allowing us to pinpoint the exact numbers that meet the required criteria. This step-by-step analysis is crucial in solving mathematical problems that involve multiple conditions.
After identifying potential solutions, it is crucial to verify that they indeed satisfy all the conditions of the problem. This step ensures that we have not made any errors in our reasoning or calculations. In this case, we have identified two potential numbers: 925 and 475. Let's verify them:
- 925: The digits of 925 are 9, 2, and 5. Their sum is 9 + 2 + 5 = 16, which satisfies the digit sum condition. The last two digits, 25, are divisible by 25, which satisfies the divisibility rule. Therefore, 925 is a valid solution.
- 475: The digits of 475 are 4, 7, and 5. Their sum is 4 + 7 + 5 = 16, which satisfies the digit sum condition. The last two digits, 75, are divisible by 25, which satisfies the divisibility rule. Therefore, 475 is also a valid solution.
By verifying the solutions, we confirm that both 925 and 475 meet the criteria of being three-digit numbers with a digit sum of 16 and being divisible by 25. This verification step is an essential part of the problem-solving process, ensuring the accuracy and reliability of our answer. It also provides confidence in our solution and confirms that we have correctly applied the given conditions.
Based on our analysis, there are two three-digit numbers whose digits sum to 16 and are divisible by 25. These numbers are 925 and 475. Therefore, the answer to the question "How many three-digit numbers, whose digits sum to 16, are divisible by 25?" is 2.
In conclusion, this problem demonstrates the importance of understanding and applying divisibility rules and digit sum properties in number theory. By systematically analyzing the conditions and narrowing down the possibilities, we were able to identify the two numbers that satisfy the given criteria. This problem-solving approach can be applied to a variety of similar mathematical challenges. The key to success lies in breaking down the problem into smaller, manageable steps and carefully considering each condition. The combination of logical reasoning and mathematical principles allows us to arrive at the correct solution. Furthermore, verifying the solutions is crucial to ensure accuracy and confidence in our answer. This exercise highlights the beauty and power of mathematical thinking in solving seemingly complex problems.