When 3³³³ is divided by 11, the remainder is

The correct option is 5.

To find the remainder when 3³³³ is divided by 11, we can use Fermat's Little Theorem.

Fermat's Little Theorem states that if $p$ is a prime number and $a$ is an integer not divisible by $p$, then:

$$a^{p-1}\equiv 1(\mathrm{mod}p)$$

Here, the divisor is $p=11$ (which is a prime number), and the base is $a=3$. Since 3 is not divisible by 11, we can apply the theorem directly:

$$3^{11-1}\equiv 1(\mathrm{mod}11)$$

This simplifies to:

$$3^{10}\equiv 1(\mathrm{mod}11)$$

Now, we can express the exponent 333 in terms of 10 to utilize this result:
333 = 10 × 33 + 3

Using the laws of exponents, we can rewrite the expression 3³³³ as:

$$3^{333}=3^{10\times 33+3}={\left(3^{10}\right)}^{33}\times 3^3$$

Taking the modulo 11 on both sides, we substitute $3^{10}\equiv 1$:

$$3^{333}\equiv {\left(1\right)}^{33}\times 3^3(\mathrm{mod}11)$$

Since $1^{33}=1$, this simplifies to:

$$3^{333}\equiv 1\times 27(\mathrm{mod}11)$$

$$3^{333}\equiv 27(\mathrm{mod}11)$$

Finally, we divide 27 by 11 and find the remainder:
27 = 11 × 2 + 5

Thus, $27\equiv 5(\mathrm{mod}11)$.

Therefore, the remainder when 3³³³ is divided by 11 is 5.