What will be the remainder when ${\left(265\right)}^{4081}+9$ is divided by 266?

The correct answer is 8.

We need to find the remainder when ${\left(265\right)}^{4081}+9$ is divided by 266.

Step 1: Express 265 in terms of 266.

Notice that:

$265=266-1$

This means that in modular arithmetic with modulus 266:

$265\equiv -1\left(\text{mod}266\right)$

Step 2: Raise both sides to the power 4081.

${\left(265\right)}^{4081}\equiv {(-1)}^{4081}\left(\text{mod}266\right)$

Since 4081 is an odd number, raising -1 to an odd power gives -1:

${(-1)}^{4081}=-1$

Therefore:

${\left(265\right)}^{4081}\equiv -1\equiv 265\left(\text{mod}266\right)$

Step 3: Add 9 to both sides.

${\left(265\right)}^{4081}+9\equiv 265+9=274\left(\text{mod}266\right)$

Step 4: Reduce 274 modulo 266.

$274=266\times 1+8$

So:

$274\equiv 8\left(\text{mod}266\right)$

Conclusion: The remainder when ${\left(265\right)}^{4081}+9$ is divided by 266 is 8.

The key insight here is recognizing that 265 = 266 - 1, which makes 265 congruent to -1 modulo 266. This transforms the problem into evaluating a simple power of -1, where the parity of the exponent (odd or even) determines the result.