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Answer :
- Apply the divisibility rule for 11: the difference between the sum of odd-placed digits and even-placed digits must be a multiple of 11.
- Let $x$ be the missing digit. Then $(1 + 2) - (7 + x)$ must be a multiple of 11.
- Simplify to get $-4 - x = 11k$ for some integer $k$.
- Solve for $x$ to find the missing digit: $x = 7$, so the missing digit is $\boxed{7}$.
### Explanation
1. Problem Analysis
We are given the number $172\square$ and we need to find the missing digit such that the number is divisible by 11. Let the missing digit be $x$. So the number is $172x$.
2. Divisibility Rule for 11
The divisibility rule for 11 states that a number is divisible by 11 if the difference between the sum of the digits in the odd places and the sum of the digits in the even places is either 0 or a multiple of 11.
3. Applying the Rule
Applying the divisibility rule to $172x$, we have $(1+2) - (7+x)$ must be divisible by 11. Simplifying this expression, we get $3 - (7+x) = 3 - 7 - x = -4 - x$.
4. Finding Possible Values
We need to find a digit $x$ such that $-4-x$ is divisible by 11. This means $-4-x$ can be $0, 11, -11, 22, -22$, and so on. Since $x$ is a digit, it must be between 0 and 9 inclusive.
5. Testing Values
If $-4-x = 0$, then $x = -4$, which is not a valid digit.
If $-4-x = 11$, then $x = -15$, which is not a valid digit.
If $-4-x = -11$, then $x = 7$, which is a valid digit.
If $-4-x = 22$, then $x = -26$, which is not a valid digit.
If $-4-x = -22$, then $x = 18$, which is not a valid digit.
6. The Missing Digit
The only valid digit we found is $x=7$. Therefore, the missing digit is 7.
7. Verification
The number is 1727. Let's check if it's divisible by 11: $1727 \div 11 = 157$. So, 1727 is divisible by 11.
8. Final Answer
Therefore, the missing digit is $\boxed{7}$.
### Examples
Understanding divisibility rules, like the one for 11, is useful in cryptography when encoding or decoding messages. For instance, ensuring that a key or a part of an encrypted message is divisible by 11 can serve as a check for errors during transmission or storage. This helps maintain the integrity of the data and ensures that the decryption process can proceed correctly.
- Let $x$ be the missing digit. Then $(1 + 2) - (7 + x)$ must be a multiple of 11.
- Simplify to get $-4 - x = 11k$ for some integer $k$.
- Solve for $x$ to find the missing digit: $x = 7$, so the missing digit is $\boxed{7}$.
### Explanation
1. Problem Analysis
We are given the number $172\square$ and we need to find the missing digit such that the number is divisible by 11. Let the missing digit be $x$. So the number is $172x$.
2. Divisibility Rule for 11
The divisibility rule for 11 states that a number is divisible by 11 if the difference between the sum of the digits in the odd places and the sum of the digits in the even places is either 0 or a multiple of 11.
3. Applying the Rule
Applying the divisibility rule to $172x$, we have $(1+2) - (7+x)$ must be divisible by 11. Simplifying this expression, we get $3 - (7+x) = 3 - 7 - x = -4 - x$.
4. Finding Possible Values
We need to find a digit $x$ such that $-4-x$ is divisible by 11. This means $-4-x$ can be $0, 11, -11, 22, -22$, and so on. Since $x$ is a digit, it must be between 0 and 9 inclusive.
5. Testing Values
If $-4-x = 0$, then $x = -4$, which is not a valid digit.
If $-4-x = 11$, then $x = -15$, which is not a valid digit.
If $-4-x = -11$, then $x = 7$, which is a valid digit.
If $-4-x = 22$, then $x = -26$, which is not a valid digit.
If $-4-x = -22$, then $x = 18$, which is not a valid digit.
6. The Missing Digit
The only valid digit we found is $x=7$. Therefore, the missing digit is 7.
7. Verification
The number is 1727. Let's check if it's divisible by 11: $1727 \div 11 = 157$. So, 1727 is divisible by 11.
8. Final Answer
Therefore, the missing digit is $\boxed{7}$.
### Examples
Understanding divisibility rules, like the one for 11, is useful in cryptography when encoding or decoding messages. For instance, ensuring that a key or a part of an encrypted message is divisible by 11 can serve as a check for errors during transmission or storage. This helps maintain the integrity of the data and ensures that the decryption process can proceed correctly.
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