Find the remainder when `(25)^(30)` is divided by 26.

Options

A. 0
B. 1
C. 25
D. 24

Answer

To find the remainder when \( (25)^{30} \) is divided by 26, we can follow these steps:

### Step 1: Rewrite the base
We can express \( 25 \) in terms of \( 26 \):
\[ 25 = 26 - 1 \]
So, we can rewrite \( (25)^{30} \) as:
\[ (25)^{30} = (26 - 1)^{30} \]

### Step 2: Apply the Binomial Theorem
Using the Binomial Theorem, we can expand \( (26 - 1)^{30} \):
\[ (26 - 1)^{30} = \sum_{k=0}^{30} \binom{30}{k} (26)^{30-k} (-1)^{k} \]
This expansion gives us a series of terms where \( \binom{30}{k} \) is the binomial coefficient.

### Step 3: Identify terms divisible by 26
When we divide this expression by 26, we need to identify which terms contribute to the remainder. The terms where \( k < 30 \) will have \( 26^{30-k} \) and will be divisible by 26. The only term that does not have a factor of 26 is when \( k = 30 \):
\[ \text{Term for } k = 30: \binom{30}{30} (26)^{0} (-1)^{30} = 1 \cdot 1 \cdot 1 = 1 \]

### Step 4: Calculate the remainder
The remainder when \( (25)^{30} \) is divided by 26 is given by the last term of the binomial expansion:
\[ \text{Remainder} = 1 \]

### Conclusion
Thus, the remainder when \( (25)^{30} \) is divided by 26 is:
\[ \boxed{1} \]

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