If `A#B=A+B+AB`. If for any A there is a number C such that `A#C=A,` then `C=?`

Answer

To solve the problem, we need to find the value of \( C \) such that for any \( A \), the equation \( A \# C = A \) holds true, where the operation \( A \# B \) is defined as:

\[ A \# B = A + B + AB \]

### Step 1: Set up the equation

We start with the equation given in the problem:

\[ A \# C = A \]

Substituting the definition of the operation \( A \# C \):

\[ A + C + AC = A \]

### Step 2: Simplify the equation

Now, we can simplify the equation by subtracting \( A \) from both sides:

\[ C + AC = 0 \]

### Step 3: Factor the equation

We can factor out \( C \) from the left side:

\[ C(1 + A) = 0 \]

### Step 4: Solve for \( C \)

For the product \( C(1 + A) = 0 \) to hold true, either \( C = 0 \) or \( 1 + A = 0 \). However, since \( A \) can be any number, \( 1 + A = 0 \) would only be true for a specific value of \( A \) (which is \( -1 \)). Therefore, the only solution that holds for any \( A \) is:

\[ C = 0 \]

### Conclusion

Thus, the value of \( C \) is:

\[ \boxed{0} \]