The digits A,B,C are such that the three digit numbers A88, 6B8, 86 C are divisible by 72 the determinant
`|{:(A,6,8),(8,B,6),(8,8,C):}|` is divisible by

Answer

To solve the problem, we need to determine the values of digits A, B, and C such that the three-digit numbers A88, 6B8, and 86C are all divisible by 72. We will also find the determinant of the matrix formed by these digits and check its divisibility.

### Step-by-Step Solution:

1. **Understanding the Divisibility by 72**:
A number is divisible by 72 if it is divisible by both 8 and 9.
- **Divisibility by 8**: The last three digits of the number must form a number that is divisible by 8.
- **Divisibility by 9**: The sum of the digits of the number must be divisible by 9.

2. **Checking A88 for Divisibility**:
- **Divisibility by 8**: The last three digits are 88, which is divisible by 8.
- **Divisibility by 9**: The sum of the digits is A + 8 + 8 = A + 16. This must be divisible by 9.

We can find suitable values for A:
- A + 16 ≡ 0 (mod 9) → A ≡ -16 ≡ 2 (mod 9)
- Possible values for A are 2 or 11 (but A must be a digit, so A = 2).

3. **Checking 6B8 for Divisibility**:
- **Divisibility by 8**: The last three digits are 6B8. We need to check values for B:
- 608 is divisible by 8.
- 618 is not divisible by 8.
- 628 is not divisible by 8.
- 638 is not divisible by 8.
- 648 is divisible by 8.
- 658 is not divisible by 8.
- 668 is not divisible by 8.
- 678 is not divisible by 8.
- 688 is not divisible by 8.
- 698 is not divisible by 8.
- Possible value for B: 0 or 4.

- **Divisibility by 9**: The sum of the digits is 6 + B + 8 = 14 + B. This must be divisible by 9.
- For B = 0: 14 + 0 = 14 (not divisible by 9).
- For B = 4: 14 + 4 = 18 (divisible by 9).

4. **Checking 86C for Divisibility**:
- **Divisibility by 8**: The last three digits are 86C. We need to check values for C:
- 860 is divisible by 8.
- 861 is not divisible by 8.
- 862 is not divisible by 8.
- 863 is not divisible by 8.
- 864 is divisible by 8.
- 865 is not divisible by 8.
- 866 is not divisible by 8.
- 867 is not divisible by 8.
- 868 is divisible by 8.
- 869 is not divisible by 8.
- Possible values for C: 0, 4, or 8.

- **Divisibility by 9**: The sum of the digits is 8 + 6 + C = 14 + C. This must be divisible by 9.
- For C = 0: 14 + 0 = 14 (not divisible by 9).
- For C = 4: 14 + 4 = 18 (divisible by 9).
- For C = 8: 14 + 8 = 22 (not divisible by 9).

5. **Summary of Values**:
- A = 2
- B = 4
- C = 4

6. **Forming the Determinant**:
The determinant to evaluate is:
\[ D = \begin{vmatrix} A & 6 & 8 \\ 8 & B & 6 \\ 8 & 8 & C \end{vmatrix} = \begin{vmatrix} 2 & 6 & 8 \\ 8 & 4 & 6 \\ 8 & 8 & 4 \end{vmatrix} \]

7. **Calculating the Determinant**:
Using the formula for the determinant of a 3x3 matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where:
- a = 2, b = 6, c = 8
- d = 8, e = 4, f = 6
- g = 8, h = 8, i = 4

Plugging in the values:
\[ D = 2(4*4 - 6*8) - 6(8*4 - 6*8) + 8(8*8 - 4*8) \]
\[ D = 2(16 - 48) - 6(32 - 48) + 8(64 - 32) \]
\[ D = 2(-32) - 6(-16) + 8(32) \]
\[ D = -64 + 96 + 256 \]
\[ D = 288 \]

8. **Final Result**:
The determinant \( D = 288 \) is divisible by 72.