Understanding the Problem

The question asks us to find the values of two single-digit integers, A and B, such that the number 2A19281870B is divisible by both 8 and 11. We will use the divisibility rules for 8 and 11 to solve this.

Applying the Divisibility Rule for 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

In the given number, 2A19281870B, the last three digits are 870B.

We need to find a single digit B (from 0 to 9) such that the number 870B is divisible by 8.

We can express 870B as $8700 + B$. To check divisibility by 8, we can look at the remainder of 8700 when divided by 8.

Calculation: $8700 \div 8$. Since $8000$ is divisible by 8, we only need to check $700 \div 8$. $700 = 8 \times 87 + 4$. So, $8700$ leaves a remainder of 4 when divided by 8. Mathematically, $8700 \equiv 4 \pmod{8}$.

Therefore, for 870B to be divisible by 8, the expression $(4 + B)$ must be divisible by 8.

Let's test the possible single-digit values for B:

From the table, we see that B must be 4 for $(4 + B)$ to be divisible by 8.

Now we know B=4, the number is 2A192818704.

Applying the Divisibility Rule for 11

A number is divisible by 11 if the alternating sum of its digits is divisible by 11. This means the difference between the sum of digits at odd positions and the sum of digits at even positions is either 0 or a multiple of 11.

Let's consider the number 2A192818704. We can assign positions starting from the left:

Number: 2 A 1 9 2 8 1 8 7 0 4

Position: 1 2 3 4 5 6 7 8 9 10 11

Sum of digits at odd positions (1, 3, 5, 7, 9, 11):

$S_{odd} = 2 + 1 + 2 + 1 + 7 + 4 = 17$

Sum of digits at even positions (2, 4, 6, 8, 10):

$S_{even} = A + 9 + 8 + 8 + 0 = A + 25$

The difference is $S_{odd} - S_{even} = 17 - (A + 25)$.

Difference = $17 - A - 25 = -8 - A$.

For the number to be divisible by 11, this difference $(-8 - A)$ must be a multiple of 11.

Since A is a single digit (0 to 9), let's find the range of possible values for $(-8 - A)$:

• If A = 0, the difference is $-8 - 0 = -8$.
• If A = 9, the difference is $-8 - 9 = -17$.

So, the difference $(-8 - A)$ must be between -17 and -8.

The multiples of 11 are ..., -22, -11, 0, 11, 22, ...

The only multiple of 11 within the range [-17, -8] is -11.

Therefore, we set the difference equal to -11:

$-8 - A = -11$

Solving for A:

$A = -8 + 11$

$A = 3$

So, A must be 3.

Final Answer

We have determined that for the number 2A19281870B to be divisible by 8, B must equal 4.

Subsequently, for the number 2A192818704 to be divisible by 11, A must equal 3.

Therefore, the values are A=3 and B=4.