Introduction

This article explores the intriguing problem of finding a 3-digit number that is precisely 80 times the sum of its digits. We delve into the algebraic process of solving this equation, providing a detailed step-by-step explanation to aid in understanding.

Formulating the Equation

Let the 3-digit number be represented as xyz where x, y, and z are its digits. The numerical value of the number can be expressed as:

10 10y z

The sum of its digits is:

x y z

According to the problem, the number is 80 times the sum of its digits:

10 10y z 80(x y z)

Step-by-Step Solution

First, we expand and rearrange the equation:

10 10y z 8 80y 80z

Rearranging the terms, we get:

10 10y z - 8 - 80y - 80z 0

This simplifies to:

2 - 70y - 79z 0

We rearrange it to:

2 70y 79z

Solving for the Digits

Let's consider the possible values for z first. Since z must be an integer, 79z must also be an integer. Therefore, z must be a multiple of 5. The possible values for z are 0 or 5.

Case 1: z 0

Substituting z 0 into the equation:

2 70y

This simplifies to:

x 3.5y

Since x must be an integer, y must be even. The possible values for y (0 to 9) that are even are 0, 2, 4, 6, and 8.

If y 0: x 0 - this is not a 3-digit number. If y 2: x 7 - this gives the number 720. If y 4: x 14 - not valid. If y 6: x 21 - not valid. If y 8: x 28 - not valid.

Therefore, the only valid number in this case is 720.

Case 2: z 5

Substituting z 5 into the equation:

2 70y 395

Rearranging gives:

x (70y 395)/4

To ensure x is an integer, 70y 395 must be divisible by 4. We can check even values of y (0 to 9):

If y 0: x 395/4 98.75 - not valid. If y 1: x 409/4 102.25 - not valid. If y 2: x 423/4 105.75 - not valid. If y 3: x 437/4 109.25 - not valid. If y 4: x 451/4 112.75 - not valid. If y 5: x 465/4 116.25 - not valid. If y 6: x 479/4 119.75 - not valid. If y 7: x 493/4 123.25 - not valid. If y 8: x 507/4 126.75 - not valid. If y 9: x 521/4 130.25 - not valid.

There are no valid solutions in this case.

Conclusion

The only 3-digit number that satisfies the condition is:

720

This problem showcases the importance of algebraic manipulation and the constraints of integer solutions in number theory.