In a three - digit number ABC, where A, B and C represent digits from 0 to 9, the value of the digit A equals the cube of the digits C. How many such three - digits numbers ABC exist ?

Options

A. 5
B. 10
C. 20
D. 30

Answer

To solve the problem step by step, we will analyze the conditions given for the three-digit number ABC, where A, B, and C are digits from 0 to 9, and A equals C cubed.

### Step 1: Understand the relationship between A and C
We know that:
\[ A = C^3 \]
Since A is a digit, it must be between 0 and 9 (inclusive).

### Step 2: Determine possible values for C
Next, we will find the values of C that satisfy the equation \( A = C^3 \) while ensuring that A remains a single-digit number (0 to 9).

- If \( C = 0 \):
\[ A = 0^3 = 0 \]

- If \( C = 1 \):
\[ A = 1^3 = 1 \]

- If \( C = 2 \):
\[ A = 2^3 = 8 \]

- If \( C = 3 \):
\[ A = 3^3 = 27 \] (not valid since A must be a single digit)

Since \( C \) can only be 0, 1, or 2 for \( A \) to remain a digit, we have the following valid pairs:
- \( (C = 0, A = 0) \)
- \( (C = 1, A = 1) \)
- \( (C = 2, A = 8) \)

### Step 3: Exclude invalid cases for a three-digit number
The problem specifies that ABC must be a three-digit number. Therefore, A cannot be 0, as that would make it a two-digit number. Thus, we exclude the pair \( (C = 0, A = 0) \).

We are left with:
- \( (C = 1, A = 1) \)
- \( (C = 2, A = 8) \)

### Step 4: Determine possible values for B
For both valid pairs, B can take any digit from 0 to 9 (inclusive).

- For \( (A = 1, C = 1) \):
- B can be: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 options)

- For \( (A = 8, C = 2) \):
- B can also be: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 options)

### Step 5: Calculate total combinations
Now we can calculate the total number of valid three-digit numbers:
- From \( (A = 1, C = 1) \): 10 combinations (for B)
- From \( (A = 8, C = 2) \): 10 combinations (for B)

Thus, the total number of valid three-digit numbers ABC is:
\[ 10 + 10 = 20 \]

### Final Answer
Therefore, the total number of three-digit numbers ABC that satisfy the given condition is **20**.

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