If `F(x)={((x-10[(x)/(10)]).10^([log_(10)x])+F([(x)/(10)])",","if "x ne 0),(0",","if "x=0):}`
where [ x ] stands for the greatest integer not exceeding ' x ', then F(7752) =

Options

A. 2222
B. 7777
C. 7752
D. 2775

Answer

To solve the problem, we need to evaluate the function \( F(x) \) defined as follows:

\[ F(x) = \begin{cases} x - 10 \left[ \frac{x}{10} \right] \cdot 10^{\log_{10} x} + F\left[ \frac{x}{10} \right] & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \]

We are tasked with finding \( F(7752) \).

### Step-by-Step Solution:

1. **Identify the case for \( F(7752) \)**:
Since \( 7752 \neq 0 \), we use the first case of the function definition.

\[ F(7752) = 7752 - 10 \left[ \frac{7752}{10} \right] \cdot 10^{\log_{10} 7752} + F\left[ \frac{7752}{10} \right] \]

2. **Calculate \( \left[ \frac{7752}{10} \right] \)**:
\[ \frac{7752}{10} = 775.2 \implies \left[ 775.2 \right] = 775 \]

3. **Calculate \( 10^{\log_{10} 7752} \)**:
By the properties of logarithms, \( 10^{\log_{10} x} = x \).
\[ 10^{\log_{10} 7752} = 7752 \]

4. **Substitute the values back into the function**:
\[ F(7752) = 7752 - 10 \cdot 775 \cdot 7752 + F(775) \]

5. **Calculate \( 10 \cdot 775 \cdot 7752 \)**:
\[ 10 \cdot 775 \cdot 7752 = 77500 \cdot 7752 \]

6. **Calculate \( F(775) \)**:
Since \( 775 \neq 0 \):
\[ F(775) = 775 - 10 \left[ \frac{775}{10} \right] \cdot 10^{\log_{10} 775} + F\left[ \frac{775}{10} \right] \]
Calculate \( \left[ \frac{775}{10} \right] \):
\[ \frac{775}{10} = 77.5 \implies \left[ 77.5 \right] = 77 \]
Calculate \( 10^{\log_{10} 775} \):
\[ 10^{\log_{10} 775} = 775 \]
Substitute back:
\[ F(775) = 775 - 10 \cdot 77 \cdot 775 + F(77) \]

7. **Calculate \( F(77) \)**:
Since \( 77 \neq 0 \):
\[ F(77) = 77 - 10 \left[ \frac{77}{10} \right] \cdot 10^{\log_{10} 77} + F\left[ \frac{77}{10} \right] \]
Calculate \( \left[ \frac{77}{10} \right] \):
\[ \frac{77}{10} = 7.7 \implies \left[ 7.7 \right] = 7 \]
Calculate \( 10^{\log_{10} 77} \):
\[ 10^{\log_{10} 77} = 77 \]
Substitute back:
\[ F(77) = 77 - 10 \cdot 7 \cdot 77 + F(7) \]

8. **Calculate \( F(7) \)**:
Since \( 7 \neq 0 \):
\[ F(7) = 7 - 10 \left[ \frac{7}{10} \right] \cdot 10^{\log_{10} 7} + F(0) \]
Calculate \( \left[ \frac{7}{10} \right] \):
\[ \frac{7}{10} = 0.7 \implies \left[ 0.7 \right] = 0 \]
Calculate \( 10^{\log_{10} 7} \):
\[ 10^{\log_{10} 7} = 7 \]
Substitute back:
\[ F(7) = 7 - 0 + 0 = 7 \]

9. **Substituting back to find \( F(77) \)**:
\[ F(77) = 77 - 10 \cdot 7 \cdot 77 + 7 = 77 - 770 + 7 = -686 \]

10. **Substituting back to find \( F(775) \)**:
\[ F(775) = 775 - 10 \cdot 77 \cdot 775 - 686 \]
Calculate \( 10 \cdot 77 \cdot 775 \):
\[ 10 \cdot 77 \cdot 775 = 59750 \]
Substitute back:
\[ F(775) = 775 - 59750 - 686 = -58661 \]

11. **Finally, substitute back to find \( F(7752) \)**:
\[ F(7752) = 7752 - 59750 - 58661 \]
Calculate:
\[ F(7752) = 7752 - 59750 - 58661 = -115659 \]

Thus, the final answer is:

\[ \boxed{-115659} \]