Question

What is the remainder when ${17^{200}}$ is divided by $18$ ?
(A) $1$
(B) $2$
(C) $3$
(D) $4$

Answer 1

Hint: The first question arises to us is what is a remainder? In mathematics, a remainder is the number left over after performing a certain mathematical operation. Normally, we use the term remainder in cases of division. For example, let us divide $10$ bananas among $9$ students. In that case each of the children will get $1$ banana each and $1$ banana will be left over. Therefore, in this calculation, the remaining term we obtain is $1$ i.e. $1$ is the remainder.

Complete step-by-step solution:
 Let us take another example. If we are given the task to divide the integer $4$ by the integer $3$, what do we get? We proceed with $\dfrac{4}{3}$. We know the result of the division i.e. the quotient is always 1. But still $3 \times 1 = 3$ but the divisible integer we have is 4, then there still remains a $(4 - 3) = 1$ . Therefore 1 is the required remainder in this case.
Now let us focus on the given problem. We are told to divide ${17^{200}}$ by $18$ and then find the remainder of this division.
First of all, we all know that $17 < 18$ and also $17 = 18 - 1$. So we divide ${\left( {18 - 1} \right)^{200}}$ by $18$ .
Now using the basic algebra, if we divide $x - 1$ by $x$ then definitely we get the remainder $1$ .
Again, dividing ${(x - 1)^2}$ by $x$ we get the remainder ${( - 1)^2} = 1$ .
Similarly, when we divide ${(x - 1)^3}$ by $x$, the remainder we get is ${\left( { - 1} \right)^3} = - 1$ .
Using the principle of Mathematical Induction, we therefore get the remainder when ${(x - 1)^n}$ is divided by $x$ is ${\left( { - 1} \right)^n}$.
Now basically, when $n$ is an even integer, ${\left( { - 1} \right)^n} = 1$and when $n$ is an odd integer, then ${\left( { - 1} \right)^n} = - 1$ .
Now coming to the current problem, if we divide ${17^{200}}$ by $18$ i.e., we divide ${\left( {18 - 1} \right)^{200}}$ by $18$, the required reminder we have is ${\left( { - 1} \right)^{200}} = 1$, followed by the previous lemma obtained using mathematical induction.
Hence the required remainder is $1$ .

Note: Students need to remember, this process is only valid when the form of division is generally: divide ${\left( {x - 1} \right)^n}$ by $x$. We have other ways to calculate the remainder for other types of problems. Note that, here $x,n$ are all integers. A remainder is not only obtained when an integer is divided by an integer, but also when we divide a polynomial function by another polynomial. In these cases we have the remainder theorem and in higher mathematics, the Chinese Remainder Theorem to help us out.