Find the digit at the unit's place in the number `17^1995 + 11^1995-7^1995`
` ` a. 0`
` b.1`
` c.2 `
` d.3`

Options

A. 0
B. 1
C. 2
D. 3

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: Identify the unit digits of each term
We need to find the unit digits of \( 17^{1995} \), \( 11^{1995} \), and \( 7^{1995} \).

- The unit digit of \( 17^{1995} \) is the same as the unit digit of \( 7^{1995} \) (since \( 17 \equiv 7 \mod 10 \)).
- The unit digit of \( 11^{1995} \) is the same as the unit digit of \( 1^{1995} \) (since \( 11 \equiv 1 \mod 10 \)).
- The unit digit of \( 7^{1995} \) needs to be calculated based on the pattern of unit digits in powers of 7.

### Step 2: Calculate the unit digit of \( 7^{1995} \)
The unit digits of powers of 7 follow a pattern:
- \( 7^1 \) has a unit digit of 7
- \( 7^2 \) has a unit digit of 9
- \( 7^3 \) has a unit digit of 3
- \( 7^4 \) has a unit digit of 1
- Then the pattern repeats every 4 terms: 7, 9, 3, 1.

To find the unit 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 digit of \( 7^{1995} \) corresponds to the unit digit of \( 7^3 \), which is 3.

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

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

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

### Step 5: Calculate the final unit digit
\[ 3 + 1 - 3 = 1 \]

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

### Final Answer
The answer is **b. 1**.