Question

When ${{2}^{256}}$ is divided by $17$, what will be the remainder?
A) $1$
B) $16$
C) $14$
D) None of these

Answer 1

We have to remind that ${{2}^{4}}$ is equal to $16$ and $16$ is only one less than $17$. So the remainder of ${{2}^{256}}$ is equal to left powers the remainder of ${{2}^{4}}$ when it is divided by $17$. We get the answer without dividing the actual number.

Complete step by step solution:
Given dividend is ${{2}^{256}}$ and divisor is $17$
Then, $Q+r=\dfrac{{{2}^{256}}}{17}$
Where $Q$ is quotient, and $r$ is remainder.
$\Rightarrow {{2}^{256}}={{\left( {{2}^{4}} \right)}^{64}}$
$\Rightarrow {{2}^{256}}={{\left( 16 \right)}^{64}}$
If we divide $16$ by $17$ then we get a remainder $16$ that we also call $-1$ .
So $r={{\left( -1 \right)}^{64}}$
If power is even then,
$\Rightarrow {{\left( -1 \right)}^{64}}=1$
So the remainder will be $1$.

Hence (A) option is correct.

Additional Information:
In division we will see the relationship between the dividend, divisor, quotient and remainder. The number which we divide is called the dividend. The number by which we divide is called the divisor. The result obtained is called the quotient. The number left over is called the remainder.
Dividend = divisor × quotient + remainder

Note: Don’t confuse over dividend and divisor, remainder for any dividend is not calculated by this method, it should be less than divisor. This method only works when the remainder is one to power something. Remainder is never bigger than dividend.