Solving the Puzzle of a Two-Digit Number That is Three Times the Sum of Its Digits

Have you ever pondered over a two-digit number and marveled at its properties? In this article, we will dive into a fascinating mathematical puzzle: finding a two-digit number that is three times the sum of its digits. We will explore the algebraic process to solve this intriguing problem and discover why the number 27 stands out in this context.

The Algebraic Approach

To understand the problem, let's define the two-digit number as 10a b, where a represents the tens digit and b is the units digit. The number is given by the condition: 10a b 3(a b). Let's simplify this equation step by step:

Starting with the given equation: 10a b 3(a b) Expanding the right side: 10a b 3a 3b Rearranging terms to isolate variables involving a and b: 10a b - 3a - 3b 0 Simplifying the equation: 7a - 2b 0 Expressing b in terms of a: 2b 7a b ( frac{7a}{2} )

Since b must be a digit (0-9), 7a must be an even number, implying that a must be even. Therefore, the possible even digits for a (the tens digit) are 2, 4, 6, and 8. Let's substitute these values into the equation for b:

If a 2, then b ( frac{7 times 2}{2} 7 ) → Number is 27 If a 4, then b ( frac{7 times 4}{2} 14 ) → Not a valid digit for b (14 is a two-digit number) If a 6, then b ( frac{7 times 6}{2} 21 ) → Not a valid digit for b (21 is a two-digit number) If a 8, then b ( frac{7 times 8}{2} 28 ) → Not a valid digit for b (28 is a two-digit number)

The only valid two-digit number is 27. Let's verify:

The sum of the digits 2 7 9 Three times the sum is 3 × 9 27 Thus, 27 satisfies the given condition.

Further Exploration

Now that we have found the solution, let's consider another condition: adding 48 to 27 and then reversing the digits of the result. We add 48 to 27:

27   48  75

Next, we reverse the digits of 75:

75 - 57

Since 57 is not the digit-reverse of 27, we conclude that there is no second valid number that meets both conditions.

The unique solution to our problem is 27.

Conclusion

Through a detailed algebraic approach, we have determined that the only two-digit number which is three times the sum of its digits is 27. This problem showcases the elegance and depth of number theory and highlights the importance of systematic problem-solving techniques in mathematics.