Discover the Positive Integer with a Tenth Digit 5 More Than Its Unit Digit and a Sum of Digits 11
In the realm of number theory, one fascinating challenge is to find a positive integer that satisfies specific conditions. Letrsquo;s explore an intriguing problem: what is a positive integer whose tenth digit is 5 more than its unit digit and the sum of its digits is 11? This article delves into the method to solve this problem, discusses various interpretations, and highlights the significance of precision in mathematical communication.
Understanding the Problem
The problem statement requires us to identify a positive integer where the tenth digit is 5 more than the unit digit, and the sum of all its digits equals 11. Letrsquo;s break down the steps to solve this.
Step 1: Define the Digits
Let us use algebra to define the digits. Let the unit digit be denoted by u, and the tenth digit by t. Given that the tenth digit is 5 more than the unit digit:
t u 5
Step 2: Sum of Digits
According to the problem, the sum of the digits is 11. Therefore, we have:
t u 11
Step 3: Solve for the Digits
Substitute the expression for t into the sum of digits equation:
u 5 u 11
Combine like terms:
2u 5 11
Subtract 5 from both sides:
2u 6
Divide both sides by 2:
u 3
Substitute u back into the expression for t:
t 3 5 8
Step 4: Construct the Integer
The integer can be formed by placing these digits in the appropriate positions. In a two-digit number, the integer would be:
10t u 10 middot; 8 3 80 3 83
Thus, the positive integer is 83.
Verification
To verify, the unit digit is 3, and the tenth digit is 8. The tenth digit is indeed 5 more than the unit digit. The sum of the digits is 8 3 11, which matches the condition. Hence, 83 is confirmed as the correct integer.
Alternative Interpretations and Length Considerations
Itrsquo;s important to consider the potential alternative interpretations and the length of the number. Some might argue that the question could refer to a number with more than two digits, where the tenth digit (not the tens digit) is 5 more than the unit digit.
Examples with More Digits
Consider numbers with a different number of digits, where the tenth digit is 6 and the unit digit is 1:
6000000041, 6000000401, 6000004001, 6000040001, 6000400001, 6004000001, 6040000001, 6400000001
In each case, the tenth digit is 6, which is 5 more than the unit digit 1, and the sum of the digits is 11. However, the question specifically referred to the tenth digit, not the tens digit.
The Importance of Clarity in Communication
Careful examination of the questionrsquo;s wording is crucial. If the tenth digit (not the tens digit) is 5 more than the unit digit, then numbers with more digits that meet this condition can be valid solutions. However, based on the provided phraseology, the integer 83 is the unique solution for a two-digit number.
Conclusion
Through careful analysis and understanding of the problemrsquo;s details, we can confidently identify the positive integer that meets the conditions. Whether we are working with a two-digit number or exploring longer numbers, precision in mathematical communication and clear problem definition are key.