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The problem in the image asks to demonstrate that the determinant of a 3x3 matrix is divisible by 72, given that three numbers \( A88 \), \( 6B8 \), and \( 86C \) are divisible by 72, where \( A \), \( B \), and \( C \) are digits.

The matrix in the problem is:

\[
\begin{vmatrix}
A & 6 & 8 \\
8 & B & 6 \\
8 & 8 & C
\end{vmatrix}
\]

### Step-by-Step Solution:

#### 1. Analyze divisibility by 72
To prove divisibility by 72, we need to consider the properties of 72:
\[
72 = 8 \times 9
\]
So, the determinant must be divisible by both 8 and 9.

#### 2. Use the fact that \( A88 \), \( 6B8 \), and \( 86C \) are divisible by 72
Each of these numbers is divisible by 72, meaning:
- \( A88 \equiv 0 \pmod{72} \)
- \( 6B8 \equiv 0 \pmod{72} \)
- \( 86C \equiv 0 \pmod{72} \)

We’ll use this fact to infer that these digits satisfy both:
- Divisibility by 8
- Divisibility by 9

#### 3. Divisibility by 8
For a number to be divisible by 8, the last three digits must form a number divisible by 8. Therefore:
- \( A88 \equiv 0 \pmod{8} \)
- \( 6B8 \equiv 0 \pmod{8} \)
- \( 86C \equiv 0 \pmod{8} \)

This gives us three congruences that restrict the values of \( A \), \( B \), and \( C \) modulo 8.

#### 4. Divisibility by 9
For a number to be divisible by 9, the sum of its digits must be divisible by 9. Therefore:
- \( A + 8 + 8 \equiv 0 \pmod{9} \)
- \( 6 + B + 8 \equiv 0 \pmod{9} \)
- \( 8 + 6 + C \equiv 0 \pmod{9} \)

These congruences provide additional restrictions on the values of \( A \), \( B \), and \( C \).

#### 5. Determinant of the matrix
The determinant of the 3x3 matrix can be computed using the formula for a 3x3 determinant:

\[
\det = A \begin{vmatrix} B & 6 \\ 8 & C \end{vmatrix} - 6 \begin{vmatrix} 8 & 6 \\ 8 & C \end{vmatrix} + 8 \begin{vmatrix} 8 & B \\ 8 & 8 \end{vmatrix}
\]

Calculating each of the 2x2 minors:

\[
\begin{vmatrix} B & 6 \\ 8 & C \end{vmatrix} = BC - 48
\]
\[
\begin{vmatrix} 8 & 6 \\ 8 & C \end{vmatrix} = 8C - 48
\]
\[
\begin{vmatrix} 8 & B \\ 8 & 8 \end{vmatrix} = 64 - 8B
\]

Substitute these into the determinant formula:

\[
\det = A(BC - 48) - 6(8C - 48) + 8(64 - 8B)
\]
Simplifying:

\[
\det = ABC - 48A - 48C + 288 + 512 - 64B
\]
\[
\det = ABC - 48A - 48C + 512 - 

Considere tres dígitos A, B, y C tales que los números A88, 6B8, y 86C son todos divisibles por 72. Sin hallar A, B, ni C, demuestre que la matriz |A 6 8| |8 B 6| |8 8 C| es divisible por 72.

Learn how to demonstrate that the determinant of a 3x3 matrix is divisible by 72, given that specific numbers A88, 6B8, and 86C are divisible by 72. This problem involves matrix determinants, divisibility rules, and number theory concepts, suitable for undergraduate-level students.

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