Question

If $2^{35}$ is divided by 5 then what will be the remainder ?

Answer 1

Hint:A polynomial is an algebraic expression with one or more terms in which an addition or a subtraction sign separates a constant and a variable. The remainder theorem is useful because it helps us find the remainder without the actual polynomial division. The laws of exponents simplify the multiplication and division operations and help to solve the problems easily.

Complete step by step answer:
We know that, a remainder is an amount left over after division (happens when the first number does not divide exactly by the other). The remainder is always less than the divisor.
Given that, \[\dfrac{{{2^{35}}}}{5}\]. The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers.We will use the cycle method of the remainder theorem as below:
\[\dfrac{{{2^1}}}{5} = 2\]
\[\Rightarrow \dfrac{{{2^2}}}{5} = 4\]
\[\Rightarrow \dfrac{{{2^3}}}{5} = 3\]
\[\Rightarrow \dfrac{{{2^4}}}{5} = 1\]
And so on….

We will use, smallest possible number greater than the divisor (here, it is 5).
Here, we will have the above expression as below:
\[\dfrac{{{2^{32}} \times {2^3}}}{5}\]
Here, we will split the number 32 as below and we will get,
\[\dfrac{{{{\left( {{2^4}} \right)}^8} \times {2^3}}}{5}\]
By using cycle method, we will get,
\[\dfrac{{{1^8} \times {2^3}}}{5}\]
On simplifying this, we will get,
\[\dfrac{{1 \times 8}}{5}= \dfrac{8}{5} = 3\]

Another Method:
\[{2^{35}} \div 5\]
We will split number 35 as below:
\[{2^{(4 \times 8 + 3)}} \div 5\]
\[\Rightarrow {2^{(4 \times 8)}} \times {2^3} \div 5\]
Removing the brackets of the power, we will get,
\[{16^8} \times 8 \div 5\]
Here, \[{16^8} \div 5\] for that, we get the remainder as 1.
And for, \[8 \div 5\] for that, we get the remainder as 3.
Thus, the remainder is \[1 \times 3 = 3\].

Hence, the remainder for \[\dfrac{{{2^{35}}}}{5} = 3\].

Note:As you can see in solving this example, we have used the concept of negative remainder. In some cases, using the negative remainder can reduce your calculations significantly. Thus, exponents or powers denote the number of times a number can be multiplied. The remainder is left over when a few things are divided into groups with an equal number of things.