`(2^(111))/(5)` find R.

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the remainder \( R \) when \( \frac{2^{111}}{5} \), we can follow these steps:

### Step 1: Understand the Cyclicity of Powers of 2
The powers of 2 have a cyclic pattern in their last digits:
- \( 2^1 = 2 \)
- \( 2^2 = 4 \)
- \( 2^3 = 8 \)
- \( 2^4 = 16 \) (last digit is 6)
- \( 2^5 = 32 \) (last digit is 2)

The last digits repeat every 4 terms: 2, 4, 8, 6.

### Step 2: Determine the Relevant Power
Since the cycle repeats every 4, we need to find \( 111 \mod 4 \):
\[ 111 \div 4 = 27 \quad \text{(remainder 3)} \]
Thus, \( 111 \mod 4 = 3 \).

### Step 3: Find the Last Digit of \( 2^{111} \)
From our cyclicity, \( 2^{111} \) has the same last digit as \( 2^3 \):
\[ 2^3 = 8 \]

### Step 4: Calculate \( 2^{111} \mod 5 \)
Now we need to find \( 2^{111} \mod 5 \). We can use the fact that \( 2^4 \equiv 1 \mod 5 \) (since \( 16 \mod 5 = 1 \)).

Now, we calculate \( 111 \mod 4 \):
\[ 111 \div 4 = 27 \quad \text{(remainder 3)} \]
Thus, \( 111 \mod 4 = 3 \), so:
\[ 2^{111} \equiv 2^3 \mod 5 \]
Calculating \( 2^3 \):
\[ 2^3 = 8 \]
Now, find \( 8 \mod 5 \):
\[ 8 \div 5 = 1 \quad \text{(remainder 3)} \]
So, \( 2^{111} \mod 5 = 3 \).

### Step 5: Conclusion
The remainder \( R \) when \( 2^{111} \) is divided by 5 is:
\[ R = 3 \]

### Final Answer
The remainder \( R \) is \( 3 \).

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