Three-Digit Numbers Divisible By 25 With Digit Sum 16

Introduction

In the realm of mathematical problem-solving, certain questions stand out for their elegance and the way they challenge our understanding of number theory. This article delves into one such intriguing problem: determining the count of three-digit numbers that meet two specific criteria. First, the sum of their digits must equal 16. Second, these numbers must be divisible by 25. This seemingly straightforward question opens the door to a fascinating exploration of divisibility rules, digit manipulation, and systematic problem-solving techniques. We will dissect the problem, employ logical reasoning, and arrive at the solution, providing a comprehensive understanding of the underlying concepts. This article is designed for students, math enthusiasts, and anyone looking to sharpen their problem-solving skills. By the end of this exploration, you'll not only have the answer but also a refined approach to tackling similar mathematical challenges.

Understanding the Problem

To effectively tackle the problem of finding three-digit numbers divisible by 25 with a digit sum of 16, we need to break it down into manageable components. First, let's clarify the conditions: we are looking for numbers in the range of 100 to 999 (three-digit numbers). Each of these numbers must satisfy two conditions. The sum of its digits must be exactly 16, and the number itself must be divisible by 25. This divisibility rule is crucial as it significantly narrows down the possibilities. A number is divisible by 25 if its last two digits are 00, 25, 50, or 75. This is a fundamental concept in number theory and a key to solving this problem efficiently. Once we understand these basic principles, we can start exploring the possible combinations of digits that meet both conditions. We will systematically analyze the possible endings (00, 25, 50, 75) and then determine which of these can have a first digit that results in a total digit sum of 16. By carefully considering these constraints, we can avoid unnecessary calculations and focus on the most promising candidates. This initial phase of understanding the problem is critical for developing a clear strategy and ultimately arriving at the correct solution.

Divisibility Rule of 25 and Its Implications

The divisibility rule of 25 serves as the cornerstone of our approach to solving this problem. A number is divisible by 25 if and only if its last two digits are 00, 25, 50, or 75. This rule stems from the fact that 100 is divisible by 25, and any number can be expressed as a multiple of 100 plus its last two digits. Therefore, the divisibility by 25 depends solely on the last two digits. In our context, this rule immediately restricts the possible three-digit numbers we need to consider. Instead of examining all numbers between 100 and 999, we only need to focus on those ending in 00, 25, 50, or 75. This significantly reduces the scope of our search and makes the problem more manageable. For example, numbers like 125, 350, and 975 are potential candidates because they end in 25, 50, and 75, respectively. However, a number like 123 can be immediately discarded because it doesn't adhere to the divisibility rule of 25. Understanding and applying this rule is not just a shortcut; it's a fundamental step in efficiently solving this type of problem. It allows us to filter out irrelevant numbers and concentrate on those that have a higher chance of meeting both conditions: divisibility by 25 and a digit sum of 16. The next step is to analyze each possible ending in conjunction with the digit sum requirement.

Case-by-Case Analysis: Possible Endings

To find the three-digit numbers that meet our criteria, we'll conduct a case-by-case analysis based on the possible endings dictated by the divisibility rule of 25: 00, 25, 50, and 75. This systematic approach ensures that we don't miss any potential solutions.

Case 1: Numbers Ending in 00

If a three-digit number ends in 00, the last two digits contribute 0 to the digit sum. This means that the first digit must be 16 to satisfy the condition that the digit sum is 16. However, since the first digit of a three-digit number can only be between 1 and 9, this case yields no solutions. A digit cannot be 16, so no number ending in 00 can have a digit sum of 16.

Case 2: Numbers Ending in 25

For numbers ending in 25, the last two digits contribute 2 + 5 = 7 to the digit sum. To reach a total digit sum of 16, the first digit must be 16 - 7 = 9. Therefore, the number 925 is a potential solution. We need to verify that 925 is indeed divisible by 25 (which it is) and that the sum of its digits is 16 (9 + 2 + 5 = 16). Thus, 925 is a valid solution.

Case 3: Numbers Ending in 50

If a three-digit number ends in 50, the last two digits contribute 5 + 0 = 5 to the digit sum. To achieve a total digit sum of 16, the first digit must be 16 - 5 = 11. Similar to the case with numbers ending in 00, the first digit cannot be 11 because it must be a single digit between 1 and 9. Therefore, there are no solutions in this case.

Case 4: Numbers Ending in 75

For numbers ending in 75, the last two digits contribute 7 + 5 = 12 to the digit sum. To reach a total of 16, the first digit must be 16 - 12 = 4. Thus, the number 475 is a potential solution. We confirm that 475 is divisible by 25 and that the sum of its digits is 4 + 7 + 5 = 16. Therefore, 475 is a valid solution.

By analyzing each case, we have identified the possible three-digit numbers that satisfy both conditions. This methodical approach ensures accuracy and completeness in our solution.

Identifying the Solutions

After our case-by-case analysis, we have pinpointed the specific three-digit numbers that are divisible by 25 and have a digit sum of 16. Our analysis revealed two such numbers:

• 925: This number ends in 25, satisfying the divisibility rule for 25. The sum of its digits is 9 + 2 + 5 = 16, meeting the second condition.
• 475: This number ends in 75, also satisfying the divisibility rule for 25. The sum of its digits is 4 + 7 + 5 = 16, fulfilling the requirement for the digit sum.

These are the only two numbers that fit both criteria. Numbers ending in 00 were ruled out because the first digit would have to be 16 to make the digit sum 16, which is impossible for a single digit. Similarly, numbers ending in 50 required the first digit to be 11, an invalid single-digit value. This systematic elimination process, combined with the divisibility rule of 25, allowed us to efficiently narrow down the possibilities and identify the solutions. Therefore, we can confidently state that there are exactly two three-digit numbers that meet the specified conditions.

Final Answer and Conclusion

Having conducted a thorough analysis, we have arrived at the final answer to our problem. There are two three-digit numbers whose digits sum to 16 and are divisible by 25. These numbers are 925 and 475. Our approach involved understanding the divisibility rule of 25, which significantly narrowed down the possible candidates. By systematically examining each case based on the possible endings (00, 25, 50, and 75), we were able to identify the numbers that met both the divisibility and digit sum criteria. This problem highlights the importance of breaking down complex mathematical questions into smaller, manageable parts. By applying logical reasoning and fundamental number theory principles, we efficiently arrived at the solution. This exercise not only answers the specific question but also reinforces the problem-solving skills applicable to a wide range of mathematical challenges. The key takeaway is the power of systematic analysis and the effective use of divisibility rules in number theory problems. We can confidently conclude that the answer is B) 2.