Find the digit at unti's place in the number `17^(1995)+11^(1995)-7^(1995).`

Answer

To find the digit at the unit's place in the expression \( 17^{1995} + 11^{1995} - 7^{1995} \), we can follow these steps:

### Step 1: Determine the unit's digits of each term
We need to find the unit's digits of \( 17^{1995} \), \( 11^{1995} \), and \( 7^{1995} \).

- The unit's digit of \( 17^{1995} \) is the same as the unit's digit of \( 7^{1995} \) (since \( 17 \equiv 7 \mod 10 \)).
- The unit's digit of \( 11^{1995} \) is the same as the unit's digit of \( 1^{1995} \) (since \( 11 \equiv 1 \mod 10 \)).
- The unit's digit of \( 7^{1995} \) needs to be calculated.

### Step 2: Find the unit's digit of \( 7^{1995} \)
The unit's digits of powers of \( 7 \) follow a pattern:
- \( 7^1 = 7 \) (unit's digit is 7)
- \( 7^2 = 49 \) (unit's digit is 9)
- \( 7^3 = 343 \) (unit's digit is 3)
- \( 7^4 = 2401 \) (unit's digit is 1)
- \( 7^5 = 16807 \) (unit's digit is 7)

The pattern repeats every 4 terms: \( 7, 9, 3, 1 \).

To find the unit's digit of \( 7^{1995} \), we calculate \( 1995 \mod 4 \):
\[ 1995 \div 4 = 498 \quad \text{remainder } 3 \]
So, \( 1995 \mod 4 = 3 \). Therefore, the unit's digit of \( 7^{1995} \) corresponds to the third term in the pattern, which is \( 3 \).

### Step 3: Find the unit's digit of \( 11^{1995} \)
Since the unit's digit of \( 11^{1995} \) is the same as \( 1^{1995} \), it is \( 1 \).

### Step 4: Combine the unit's digits
Now we have:
- Unit's digit of \( 17^{1995} \) (which is the same as \( 7^{1995} \)) = \( 3 \)
- Unit's digit of \( 11^{1995} = 1 \)
- Unit's digit of \( 7^{1995} = 3 \)

Now we substitute these into the original expression:
\[ \text{Unit's digit of } (17^{1995} + 11^{1995} - 7^{1995}) = (3 + 1 - 3) \]

### Step 5: Calculate the final unit's digit
Calculating this gives:
\[ 3 + 1 - 3 = 1 \]

Thus, the digit at the unit's place in the number \( 17^{1995} + 11^{1995} - 7^{1995} \) is \( 1 \).

### Final Answer
The unit's digit is \( 1 \).