Introduction

What is the three-digit number that is exactly equal to 18 times the sum of its digits? This problem is an engaging puzzle that involves dividing a number by 18, a composite number with prime factors of 2 and 9. By understanding divisibility rules and the characteristics of the number 18, we can find the solution to this intriguing problem.

Understanding the Problem

A three-digit number can be represented as XYZ, where X, Y, and Z are its digits. According to the problem, we need to find the number such that:

XYZ 18(X Y Z)

This equation implies that the number XYZ is divisible by 18. Since 18 2 × 9, the number must be divisible by both 2 and 9.

Divisibility by 9

For a number to be divisible by 9, the sum of its digits must be divisible by 9. This means:

X Y Z must be 9, 18, or 27.

Solving the Equation

Let's break down the problem step by step:

The sum of the digits X, Y, and Z must equal 9, 18, or 27. The number must be 18 times this sum. Let's start with the smallest sum, 9.

Step-by-Step Solution

Sum 9:

18 × 9 162

Check the digits of 162: 1 6 2 9, which is correct.

Sum 18:

18 × 18 324

Check the digits of 324: 3 2 4 9, which is not 18.

Sum 27:

18 × 27 486

Check the digits of 486: 4 8 6 18, which is not 27.

The only valid solution from the above checks is 162.

Alternative Solution Method

We can also use algebra to solve the problem. Consider the number XYZ as 10 10Y Z. According to the problem:

10 10Y Z 18(X Y Z)

Simplify this equation:

10 10Y Z 18X 18Y 18Z

82X - 8Y 17Z

This implies that Z must be an even number (since 17Z must be even).

If Z 2, then:

82X - 8Y 34

41X - 4Y 17

From this, we can solve for Y in terms of X:

Y (41X - 17) / 4

For Y to be an integer, 41X - 17 must be divisible by 4. Testing X 1:

Y (41 - 17) / 4 24 / 4 6

Thus, the number is 101062 162.

Conclusion

The only three-digit number that satisfies the condition XYZ 18(X Y Z) is 162. This problem demonstrates the application of divisibility rules and algebraic manipulation to solve number theory problems.

Further Reading

Solving similar problems involving sums of digits and divisibility criteria can be valuable for enhancing problem-solving skills in number theory. If you are interested in exploring more such problems, consider looking at:
- Divisibility rules for other numbers
- Problems involving the sum of digits
- Number theory puzzles in competitive programming