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Answer:
Step-by-step explanation:To determine the smallest digit to replace `*` so that the number \(380 * 692\) is divisible by 11, we can use the divisibility rule for 11.
The rule states that a number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is a multiple of 11 (including 0).
Let's denote the number as \(380 * 692\). We will place the digits into their respective positions as follows:
- The odd-position digits: 3, 0, *, 6, and 2.
- The even-position digits: 8, *, 9, and 2.
First, we calculate the sum of the odd-position digits:
\[ 3 + 0 + * + 6 + 2 \]
Next, we calculate the sum of the even-position digits:
\[ 8 + * + 9 + 2 \]
Let’s simplify these sums:
\[ \text{Sum of odd-position digits} = 3 + 0 + * + 6 + 2 = 11 + * \]
\[ \text{Sum of even-position digits} = 8 + * + 9 + 2 = 19 + * \]
According to the divisibility rule for 11, the absolute difference between these sums should be a multiple of 11:
\[ |(11 + *) - (19 + *)| = |11 + * - 19 - *| = |11 - 19| = | -8 | = 8 \]
For the number to be divisible by 11, this difference must be 0 or 11, but 8 is neither 0 nor 11. Therefore, we need to make an adjustment by finding the smallest digit for `*` such that the resulting number meets the criteria of divisibility by 11.
To make this divisible by 11, we need to find an appropriate value for `*` such that the difference is 0 or 11. Since our sum of digits currently has an absolute difference of 8, let's find how we can balance the sums:
\[ \text{Sum of odd-position digits} = 11 + * \]
\[ \text{Sum of even-position digits} = 19 + * \]
If we try \( | (11 + *) - (19 + *)| \), we still get 8. We can balance this by adding and trying values for `*` as follows:
1. If we set \( * = 1 \), we get:
- Odd position sum: \(11 + 1 = 12\)
- Even position sum: \(19 + 1 = 20\)
- Difference: \( |12 - 20| = 8 \) - not a multiple of 11
2. If we set \( * = 2 \), we get:
- Odd position sum: \(11 + 2 = 13\)
- Even position sum: \(19 + 2 = 21\)
- Difference: \( |13 - 21| = 8 \) - not a multiple of 11
3. If we set \( * = 3 \), we get:
- Odd position sum: \(11 + 3 = 14\)
- Even position sum: \(19 + 3 = 22\)
- Difference: \( |14 - 22| = 8 \) - not a multiple of 11
4. If we set \( * = 4 \), we get:
- Odd position sum: \(11 + 4 = 15\)
- Even position sum: \(19 + 4 = 23\)
- Difference: \( |15 - 23| = 8 \) - not a multiple of 11
5. If we set \( * = 5 \), we get:
- Odd position sum: \(11 + 5 = 16\)
- Even position sum: \(19 + 5 = 24\)
- Difference: \( |16 - 24| = 8 \) - not a multiple of 11
6. If we set \( * = 6 \), we get:
- Odd position sum: \(11 + 6 = 17\)
- Even position sum: \(19 + 6 = 25\)
- Difference: \( |17 - 25| = 8 \) - not a multiple of 11
7. If we set \( * = 7 \), we get:
- Odd position sum: \(11 + 7 = 18\)
- Even position sum: \(19 + 7 = 26\)
- Difference: \( |18 - 26| = 8 \) - not a multiple of 11
8. If we set \( * = 8 \), we get:
- Odd position sum: \(11 + 8 = 19\)
- Even position sum: \(19 + 8 = 27\)
- Difference: \( |19 - 27| = 8 \) - not a multiple of 11
9. If we set \( * = 9 \), we get:
- Odd position sum: \(11 + 9 = 20\)
- Even position sum: \(19 + 9 = 28\)
- Difference: \( |20 - 28| = 8 \) - not a multiple of 11
As we see, by using these calculations for `* = 1` through `* = 9`, none of these numbers are divisible by 11.
Thus, based on the strict calculation above, it appears that for \( * \) to achieve divisibility by 11 under normal interpretations is impossible. The initial criteria themselves might have been misunderstood or there was a calculation error in the problem constraint. We’ll need more information to fix this error.
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