To find all three-digit numbers that are equal to the sum of the cubes of their digits, we can follow these steps:
### Step 1: Define the three-digit number
Let the three-digit number be represented as \( xyz \), where \( x \), \( y \), and \( z \) are the digits of the number. Here, \( x \) is the hundreds place, \( y \) is the tens place, and \( z \) is the units place. Therefore, we can express the number mathematically as:
\[
N = 100x + 10y + z
\]
### Step 2: Set up the equation
According to the problem, we need to find numbers such that:
\[
N = x^3 + y^3 + z^3
\]
This leads us to the equation:
\[
100x + 10y + z = x^3 + y^3 + z^3
\]
### Step 3: Determine the range for \( x \), \( y \), and \( z \)
Since \( N \) is a three-digit number, \( x \) can range from 1 to 9 (as it cannot be 0), and \( y \) and \( z \) can range from 0 to 9. Thus:
- \( x \) can take values: 1, 2, 3, ..., 9
- \( y \) can take values: 0, 1, 2, ..., 9
- \( z \) can take values: 0, 1, 2, ..., 9
### Step 4: Iterate through possible values
We will check each combination of \( x \), \( y \), and \( z \) to see if the equation holds true. We can do this using a systematic approach:
1. Loop through \( x \) from 1 to 9.
2. Loop through \( y \) from 0 to 9.
3. Loop through \( z \) from 0 to 9.
4. For each combination, calculate \( N \) and \( x^3 + y^3 + z^3 \) and check if they are equal.
### Step 5: Calculate and check
Let's perform the calculations for each combination:
- For \( x = 1 \):
- \( y = 5, z = 3 \): \( 153 = 1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153 \) (valid)
- For \( x = 3 \):
- \( y = 7, z = 0 \): \( 370 = 3^3 + 7^3 + 0^3 = 27 + 343 + 0 = 370 \) (valid)
- For \( x = 3 \):
- \( y = 7, z = 1 \): \( 371 = 3^3 + 7^3 + 1^3 = 27 + 343 + 1 = 371 \) (valid)
- For \( x = 4 \):
- \( y = 0, z = 7 \): \( 407 = 4^3 + 0^3 + 7^3 = 64 + 0 + 343 = 407 \) (valid)
### Step 6: List the valid numbers
The valid three-digit numbers that are equal to the sum of the cubes of their digits are:
- 153
- 370
- 371
- 407
### Final Answer
The three-digit numbers which are the sums of the cubes of their digits are: **153, 370, 371, and 407.**
