The remainder when ('(2)^(54)' -1) is divided by 9 is :

Options

A. A)0
B. B)8
C. C)7
D. D)none of these

Answer

To find the remainder when \( 2^{54} - 1 \) is divided by 9, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression \( 2^{54} - 1 \).

### Step 2: Use the property of exponents
We can express \( 2^{54} \) as \( (2^6)^9 \). This gives us:
\[ 2^{54} - 1 = (2^6)^9 - 1^9 \]

### Step 3: Apply the difference of powers formula
Using the formula for the difference of powers, \( a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + \ldots + b^{n-1}) \), we can factor the expression:
\[ (2^6 - 1)((2^6)^{8} + (2^6)^{7} \cdot 1 + \ldots + 1^{8}) \]
Here, \( a = 2^6 \) and \( b = 1 \).

### Step 4: Calculate \( 2^6 - 1 \)
Now we calculate \( 2^6 - 1 \):
\[ 2^6 = 64 \implies 2^6 - 1 = 64 - 1 = 63 \]

### Step 5: Divide by 9
Next, we need to find the remainder of \( 63 \) when divided by \( 9 \):
\[ 63 \div 9 = 7 \quad \text{(exact division)} \]
Since \( 63 \) is divisible by \( 9 \), the remainder is \( 0 \).

### Step 6: Conclusion
Thus, the remainder when \( 2^{54} - 1 \) is divided by \( 9 \) is:
\[ \text{Remainder} = 0 \]

### Final Answer
The remainder when \( 2^{54} - 1 \) is divided by \( 9 \) is \( 0 \).

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