`(2^(112))/(9)` find R.

Answer

To find the remainder \( R \) when \( \frac{2^{112}}{9} \), we can follow these steps:

### Step 1: Identify the power of 2 in relation to 9
We start by recognizing that \( 2^3 = 8 \) which is close to 9. We can express \( 2^{112} \) in terms of powers of \( 2^3 \).

### Step 2: Rewrite the exponent
We can rewrite \( 2^{112} \) as:
\[ 2^{112} = (2^3)^{37} \cdot 2^1 \]
This is because \( 112 = 3 \times 37 + 1 \).

### Step 3: Simplify using modulo
Now, we can simplify this expression modulo 9:
\[ (2^3)^{37} \equiv 8^{37} \mod 9 \]
Since \( 8 \equiv -1 \mod 9 \), we have:
\[ 8^{37} \equiv (-1)^{37} \mod 9 \]
Since 37 is odd:
\[ (-1)^{37} \equiv -1 \mod 9 \]

### Step 4: Combine with the remaining power
Now, we need to include the remaining \( 2^1 \):
\[ 2^{112} \equiv -1 \cdot 2 \mod 9 \]
This simplifies to:
\[ -2 \mod 9 \]

### Step 5: Adjust for negative remainder
Since we cannot have a negative remainder, we adjust:
\[ -2 + 9 = 7 \]

### Conclusion
Thus, the remainder \( R \) when \( \frac{2^{112}}{9} \) is:
\[ R = 7 \]

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