Which Number Is Twice The Sum Of Its Digits

Which Number is Twice the Sum of its Digits? A Mathematical Puzzle

This intriguing mathematical puzzle asks us to find a number that's exactly double the sum of its digits. This seemingly simple problem delves into the fascinating world of number theory and offers a unique opportunity to explore different problem-solving approaches. Let's unravel the mystery together!

This article will explore various methods to solve this puzzle, from simple trial and error to more systematic approaches, ultimately leading you to the solution and a deeper understanding of number properties.

Understanding the Problem

The core of the problem lies in understanding the relationship between a number and the sum of its digits. We're looking for a number (let's call it 'x') where:

x = 2 * (sum of digits of x)

For example, let's consider the number 12. The sum of its digits is 1 + 2 = 3. Is 12 twice the sum of its digits (3)? No, 12 ≠ 2 * 3. We need to find the number that satisfies this equation.

Method 1: Trial and Error (for smaller numbers)

A straightforward, albeit less efficient approach, is to try different numbers. Start with small numbers and check if they satisfy the condition. This method is practical for smaller numbers but becomes cumbersome as the numbers increase.

Let's try a few examples:

• 1: 1 ≠ 2 * 1
• 2: 2 ≠ 2 * 2
• 3: 3 ≠ 2 * 3
• ...and so on.

While this can lead to a solution, it's not a scalable or elegant solution for larger numbers.

Method 2: A More Systematic Approach

Instead of randomly trying numbers, let's analyze the properties of such a number. Consider a two-digit number represented as 10a + b, where 'a' and 'b' are digits from 0 to 9. The equation becomes:

10a + b = 2 * (a + b)

Simplifying this equation, we get:

10a + b = 2a + 2b 8a = b

This tells us that the units digit ('b') must be eight times the tens digit ('a'). Since 'b' can't be greater than 9, the only possible solution is when 'a' = 1 and 'b' = 8.

Therefore, the number is 18. Let's verify:

The sum of digits of 18 is 1 + 8 = 9. And 2 * 9 = 18. This confirms that 18 is indeed the solution.

Extending the Search (for larger numbers)

While we've found a solution, the question remains: are there other numbers that satisfy this condition? The systematic approach helps us understand why it's unlikely to find other solutions, especially with larger numbers. The relationship between the digits severely limits the possibilities. Exploring numbers with three or more digits would require a more complex algebraic manipulation, but the underlying principle remains the same.

Conclusion

The number that is twice the sum of its digits is 18. This puzzle highlights the power of combining intuitive trial and error with a more systematic, algebraic approach to solve mathematical problems. By understanding the underlying number properties and expressing the problem algebraically, we can efficiently find the solution and explore the limitations of such number relationships.