What is the remainder when `3^(257)` is divided by 80

Answer

To find the remainder when \( 3^{257} \) is divided by 80, we can use the properties of modular arithmetic and the binomial theorem. Here’s a step-by-step solution:

### Step 1: Rewrite the exponent
We can express \( 3^{257} \) as:
\[ 3^{257} = 3^{256} \cdot 3 = 3 \cdot (3^{4})^{64} \]
This is because \( 256 = 4 \times 64 \).

### Step 2: Calculate \( 3^4 \)
Next, we calculate \( 3^4 \):
\[ 3^4 = 81 \]

### Step 3: Reduce \( 81 \) modulo \( 80 \)
Now, we find \( 81 \mod 80 \):
\[ 81 \equiv 1 \mod 80 \]

### Step 4: Substitute back into the expression
Substituting back, we have:
\[ 3^{256} = (3^4)^{64} \equiv 1^{64} \equiv 1 \mod 80 \]

### Step 5: Combine results
Now we can combine the results:
\[ 3^{257} \equiv 3 \cdot 1 \equiv 3 \mod 80 \]

### Conclusion
Thus, the remainder when \( 3^{257} \) is divided by 80 is:
\[ \boxed{3} \]

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