The digit at unit place in the number `17^(1995)+11^(1995)-7^(1995)` is

Options

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

Answer

To find the digit at the unit place in the expression \( 17^{1995} + 11^{1995} - 7^{1995} \), we will analyze the unit digits of each term separately.

### Step 1: Find the unit digit of \( 17^{1995} \)
The unit digit of \( 17^{1995} \) is the same as the unit digit of \( 7^{1995} \) because the unit digit of a number only depends on the unit digit of the base.

To determine the unit digit of \( 7^{1995} \), we observe the pattern in the unit digits of powers of 7:
- \( 7^1 = 7 \) (unit digit is 7)
- \( 7^2 = 49 \) (unit digit is 9)
- \( 7^3 = 343 \) (unit digit is 3)
- \( 7^4 = 2401 \) (unit digit is 1)
- \( 7^5 = 16807 \) (unit digit is 7)

The unit digits repeat every 4 terms: \( 7, 9, 3, 1 \).

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

### Step 2: Find the unit digit of \( 11^{1995} \)
The unit digit of \( 11^{1995} \) is the same as the unit digit of \( 1^{1995} \) since the unit digit of 11 is 1.

The unit digit of \( 1^{1995} \) is always 1.

### Step 3: Combine the unit digits
Now we can combine the unit digits we found:
- 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 calculate:
\[ \text{Unit digit of } (17^{1995} + 11^{1995} - 7^{1995}) = (3 + 1 - 3) \]
Calculating this gives:
\[ 3 + 1 = 4 \quad \text{and then} \quad 4 - 3 = 1 \]

### Final Answer
The digit at the unit place in the number \( 17^{1995} + 11^{1995} - 7^{1995} \) is **1**.

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