Question

When \[{2^{256}}\] is divided by \[17\] , what will be the remainder ?
\[\left( 1 \right)\] \[1\]
\[\left( 2 \right)\] \[16\]
\[\left( 3 \right)\] \[14\]
\[\left( 4 \right)\] None of these

Answer 1

Hint: We have to find the remainder obtained when \[{2^{256}}\] is divided by \[17\] . We solve this question using the remainder theorem of polynomials . We solve this question by relating the terms of the numerator and the denominator in terms of elements with the same powers so as to simplify the question . We would split the denominator in such a way that adding the terms gives the original number also the splitter terms can be represented in terms of power of \[2\] . The other term of the split denominator gives us the value of the remainder which we would obtain after the actual division.

Complete step-by-step solution:
Given :
\[{2^{256}}\] is divided by \[17\]
So , the required expression can also be represented as \[\dfrac{{{2^{256}}}}{{17}}\] .
Let \[f(x) = \dfrac{{{2^{256}}}}{{17}}\]
Now , simplifying the numerator we can also , write the expression as :
\[f(x) = \dfrac{{{{\left( {{2^4}} \right)}^{64}}}}{{17}}\]
[As a number \[{a^{2a}}\]can also be written as \[{\left( {{a^2}} \right)^a}\]]
Now , splitting the denominator in terms such that adding the terms gives the original number also the splitter terms can be represented in terms of power of \[2\]
\[f(x) = \dfrac{{{{\left( {2{}^4} \right)}^{64}}}}{{{2^4} + 1}}\]
The remainder theorem states that when \[f\left( y \right)\] is divided by \[\left( {y + a} \right)\] then the remainder is given as \[f\left( { - a} \right)\]
Using the theorem we compute that \[f(y) = {({2^4})^{64}}\] and \[(y + a) = {2^4} + 1\]
Also , we can write the expression as :
\[f\left( x \right) = \dfrac{{f(y)}}{{y + a}}\]
From the theorem and the considerations , we get that \[y = {2^4}\] and \[a = 1\]
Now , as we that the remainder would be given as \[f\left( { - a} \right)\]
So ,
\[remainder = f\left( { - 1} \right)\]
\[remainder = {( - 1)^{64}}\]
\[remainder = 1\]
Hence , the remainder when \[{2^{256}}\] is divided by \[17\] is \[1\] .
Thus , the correct option is \[(1)\] .

Note: This is a very conceptual question which can be solved only using the knowledge of the remainder theorem otherwise thinking of expanding the terms and then finding the remainder would not be a fruitful way of doing the question as it would be a very hard and time taking process . We simplify the terms and hence find the remainder.