The Sum Of The Digits Of A Two-digit Number Is 1/7 Of The Number. The Unit's Digit Is 4 Less Than The Ten's Digit. What Is The Remainder When The Number Obtained On Reversing Its Digits Is Divided By 7?

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In this mathematical exploration, we delve into the fascinating world of number puzzles, specifically focusing on a two-digit number with intriguing properties. This article aims to dissect the given problem, providing a step-by-step solution while elucidating the underlying mathematical concepts. Our main objective is to determine the remainder when the number formed by reversing the digits is divided by 7. This problem not only tests our understanding of place value and number theory but also our ability to translate word problems into mathematical equations. Let's embark on this journey of numerical discovery, unraveling the mystery behind this unique two-digit number and finding the ultimate remainder.

Unveiling the Two-Digit Number: A Mathematical Quest

At the heart of this puzzle lies a two-digit number, a numerical entity composed of a tens digit and a units digit. Our quest begins by deciphering the clues embedded within the problem statement. The first key piece of information reveals a unique relationship between the digits and the number itself: the sum of the digits is precisely 1/7th of the entire number. This statement subtly hints at the number's composition and provides a crucial equation for our mathematical arsenal. The second clue adds another layer of complexity, stating that the units digit lags behind the tens digit by a margin of 4. This seemingly simple phrase translates into a powerful algebraic equation, further narrowing down the possibilities. To truly grasp the essence of this problem, we must translate these verbal clues into the language of mathematics, laying the foundation for a systematic solution. Let's embark on this journey of numerical deduction, transforming words into equations and unlocking the secrets of this enigmatic two-digit number. We will meticulously analyze each clue, converting them into mathematical expressions and employing algebraic techniques to reveal the digits that form this number. This process is not merely about finding the answer; it's about understanding the underlying principles that govern number relationships and problem-solving strategies in mathematics.

Setting Up the Equations: Translating Words into Math

To effectively tackle this intriguing numerical puzzle, the initial step involves translating the verbal clues into a structured mathematical framework. Let's denote the ten's digit of our mystery number as 'x' and the unit's digit as 'y'. With this notation in place, we can express the original two-digit number as 10x + y, a direct consequence of the place value system. Now, let's dissect the first clue: "The sum of the digits of a two-digit number is 1/7 of the number." This statement can be elegantly expressed as an equation: x + y = (1/7)(10x + y). This equation encapsulates the relationship between the digits and the number itself, providing a fundamental constraint for our solution. The second clue, "The unit's digit is 4 less than the ten's digit," offers another vital piece of information. Translating this into mathematical language, we get y = x - 4. This equation establishes a direct link between the two digits, allowing us to express one in terms of the other. With these two equations in hand, we have effectively transformed the word problem into a solvable mathematical system. The next phase involves employing algebraic techniques to solve these equations, unraveling the values of x and y and ultimately revealing the digits that compose our elusive two-digit number. This meticulous translation process is not just about finding the equations; it's about understanding how mathematical notation can capture the essence of verbal descriptions, paving the way for systematic problem-solving.

Solving the Equations: Unveiling the Digits

With our equations meticulously crafted, we now embark on the crucial step of solving them to unearth the values of 'x' and 'y', the digits that hold the key to our two-digit number. Our equation system consists of two equations: x + y = (1/7)(10x + y) and y = x - 4. To solve this system, we can employ a technique called substitution, a powerful algebraic tool. Since we know that y = x - 4, we can substitute this expression for 'y' in the first equation. This substitution transforms the first equation into an equation with only one variable, 'x', making it solvable. Let's perform the substitution: x + (x - 4) = (1/7)[10x + (x - 4)]. Simplifying this equation, we get 2x - 4 = (1/7)(11x - 4). To eliminate the fraction, we can multiply both sides of the equation by 7, yielding 7(2x - 4) = 11x - 4. Expanding and simplifying further, we arrive at 14x - 28 = 11x - 4. Now, let's isolate 'x' by subtracting 11x from both sides and adding 28 to both sides. This gives us 3x = 24. Finally, dividing both sides by 3, we find that x = 8. With the value of 'x' in hand, we can easily determine 'y' using the equation y = x - 4. Substituting x = 8, we get y = 8 - 4, which gives us y = 4. Therefore, we have successfully deciphered the digits of our two-digit number: the ten's digit (x) is 8, and the unit's digit (y) is 4. This intricate process of algebraic manipulation not only reveals the solution but also showcases the power of systematic equation-solving in unraveling mathematical mysteries.

Constructing the Number and Reversing Digits

Now that we've diligently solved the equations and unveiled the digits, it's time to construct the original two-digit number and embark on the next phase of our problem: reversing the digits. Recall that we denoted the ten's digit as 'x' and the unit's digit as 'y'. We've determined that x = 8 and y = 4. Therefore, the original two-digit number is formed by placing 'x' in the tens place and 'y' in the units place, resulting in the number 84. With the original number in hand, the next step is to reverse its digits. This involves swapping the positions of the ten's and unit's digits. So, if our original number is 84, reversing the digits gives us 48. This simple yet crucial step sets the stage for the final act of our mathematical drama: finding the remainder when this reversed number is divided by 7. Constructing the number and reversing its digits is not merely a mechanical process; it's about understanding the significance of place value and how the position of a digit influences its contribution to the overall value of the number. This understanding forms the foundation for exploring further numerical relationships and solving more complex problems.

Finding the Remainder: The Final Calculation

With the reversed number, 48, in our grasp, we arrive at the final act of our mathematical quest: determining the remainder when this number is divided by 7. This step involves a fundamental arithmetic operation: division. To find the remainder, we divide 48 by 7. Performing the division, we find that 7 goes into 48 six times (7 x 6 = 42), leaving a remainder of 6 (48 - 42 = 6). Therefore, the remainder when the number obtained on reversing the digits (48) is divided by 7 is 6. This calculation not only provides the answer to our problem but also highlights the significance of remainders in number theory. Remainders play a crucial role in various mathematical concepts, including modular arithmetic and divisibility rules. Understanding how to calculate remainders is essential for solving a wide range of problems, from simple division tasks to complex number theory puzzles. This final calculation serves as a testament to the power of basic arithmetic operations in unraveling mathematical mysteries and completing our journey of numerical discovery.

In conclusion, we have successfully navigated the intricate pathways of this two-digit number puzzle, employing a combination of algebraic techniques and arithmetic operations to arrive at the solution. Our journey began with deciphering the clues embedded within the problem statement, translating them into a system of equations. We then skillfully solved these equations, unveiling the digits of our mystery number. We constructed the number, reversed its digits, and finally, calculated the remainder when the reversed number was divided by 7. The answer, 6, not only satisfies the conditions of the problem but also showcases the elegance and precision of mathematics in unraveling numerical mysteries. This problem serves as a reminder that mathematics is not just about memorizing formulas; it's about developing a logical and systematic approach to problem-solving. The skills we've honed in this exploration, such as translating word problems into equations, solving algebraic systems, and performing arithmetic operations, are invaluable tools that can be applied to a wide range of mathematical challenges. This journey of numerical discovery has not only provided us with an answer but also enriched our understanding of mathematical principles and problem-solving strategies.