Solving the Mystery of a Specific Two-Digit Number

Imagine a two-digit number with a unique characteristic: its unit digit is 3, and the sum of its digits is equivalent to 1/7 of the number itself. This intriguing puzzle involves a blend of algebra and arithmetic, leading us to uncover the elusive number. In this article, we will guide you through the detailed steps and logical reasoning involved in solving this captivating problem.

Understanding the Problem

We are tasked with finding a two-digit number, ab, where:

The unit digit is 3, hence b 3. The sum of the digits (a b) is 1/7 of the number itself.

Leveraging Mathematical Relationships

Let's represent the two-digit number as 10a 3. The sum of the digits, a 3, is equivalent to (1/7) * (10a 3). We can set up the following equation:

a 3 (1/7) * (10a 3)

To simplify, multiply both sides by 7:

7a 21 10a 3

Next, rearrange to isolate a:

21 - 3 10a - 7a

Which simplifies to:

18 3a

Finally, divide both sides by 3:

a 6

The two-digit number can now be determined as:

10a 3 10(6) 3 63

Verification and Confirmation

To ensure our solution is correct, we need to verify that the sum of the digits (6 3 9) matches 1/7 of the number itself. Indeed:

9 1/7 * 63

Thus, the number 63 satisfies all the given conditions.

Alternative Solution Techniques

For those keen on exploring alternative solution paths, another approach is to recognize that the number must be a multiple of 7 due to its defining characteristic. Checking the multiples of 7 that have a unit digit of 3, we find:

7 * 9 63

Since the sum of the digits of 63 (6 3 9) is also 1/7 of 63, 63 is the correct answer.

Conclusion

In this article, we have unraveled the mystery of a two-digit number that holds the unique property of its unit digit being 3 and the sum of its digits being 1/7 of the number itself. The solution involves algebraic manipulation, verification, and considering alternative problem-solving techniques. By understanding and applying these methods, one can efficiently tackle similar puzzles and deepen their mathematical reasoning skills.