The units digit of the square of a number and the units digits of the cube of the number are equal to the units digits of the number .How many values are possibles for the units digits of such number?

Options

A. 2
B. 4
C. 5
D. 3

Answer

To solve the problem, we need to find the units digits of numbers such that the units digit of the square of the number and the units digit of the cube of the number are equal to the units digit of the number itself. We will analyze each digit from 0 to 9.

### Step-by-Step Solution:

1. **Identify Possible Units Digits**: We will check the units digits from 0 to 9.

2. **Calculate the Square and Cube**:
- For each digit \( n \) (from 0 to 9), calculate \( n^2 \) and \( n^3 \).
- Then, find the units digit of \( n^2 \) and \( n^3 \).

3. **Compare Units Digits**:
- Check if the units digit of \( n^2 \) is equal to the units digit of \( n^3 \) and also equal to \( n \).

4. **List Valid Digits**:
- Keep track of the digits that satisfy the condition.

5. **Count Valid Digits**:
- Finally, count how many digits satisfy the condition.

### Detailed Calculations:

- **For \( n = 0 \)**:
- \( n^2 = 0^2 = 0 \) (units digit = 0)
- \( n^3 = 0^3 = 0 \) (units digit = 0)
- Condition satisfied.

- **For \( n = 1 \)**:
- \( n^2 = 1^2 = 1 \) (units digit = 1)
- \( n^3 = 1^3 = 1 \) (units digit = 1)
- Condition satisfied.

- **For \( n = 2 \)**:
- \( n^2 = 2^2 = 4 \) (units digit = 4)
- \( n^3 = 2^3 = 8 \) (units digit = 8)
- Condition not satisfied.

- **For \( n = 3 \)**:
- \( n^2 = 3^2 = 9 \) (units digit = 9)
- \( n^3 = 3^3 = 27 \) (units digit = 7)
- Condition not satisfied.

- **For \( n = 4 \)**:
- \( n^2 = 4^2 = 16 \) (units digit = 6)
- \( n^3 = 4^3 = 64 \) (units digit = 4)
- Condition not satisfied.

- **For \( n = 5 \)**:
- \( n^2 = 5^2 = 25 \) (units digit = 5)
- \( n^3 = 5^3 = 125 \) (units digit = 5)
- Condition satisfied.

- **For \( n = 6 \)**:
- \( n^2 = 6^2 = 36 \) (units digit = 6)
- \( n^3 = 6^3 = 216 \) (units digit = 6)
- Condition satisfied.

- **For \( n = 7 \)**:
- \( n^2 = 7^2 = 49 \) (units digit = 9)
- \( n^3 = 7^3 = 343 \) (units digit = 3)
- Condition not satisfied.

- **For \( n = 8 \)**:
- \( n^2 = 8^2 = 64 \) (units digit = 4)
- \( n^3 = 8^3 = 512 \) (units digit = 2)
- Condition not satisfied.

- **For \( n = 9 \)**:
- \( n^2 = 9^2 = 81 \) (units digit = 1)
- \( n^3 = 9^3 = 729 \) (units digit = 9)
- Condition not satisfied.

### Summary of Valid Units Digits:
- The valid units digits are: 0, 1, 5, and 6.

### Final Count:
- There are **4 valid units digits** (0, 1, 5, 6).

### Answer:
The total number of possible values for the units digits of such numbers is **4**.