Question

The quotient and the remainder when ${{x}^{2002}}-2001$ is divided by ${{x}^{91}}$ are respectively
(A) \[{{x}^{91\times 22}}\], 2001
(B) ${{x}^{91}}$, 2001
(C) ${{x}^{91\times 21}}$, -2001
(D) ${{x}^{9}}$, -2001

Answer 1

Hint:We solve this question by first considering ${{x}^{2002}}-2001$ and dividing it with ${{x}^{91}}$. Then we divide it until we cannot reduce it and then we can get the quotient and remainder of the division. Then we write the quotient in terms of the power of x is the product of 91 to change it as in the options given.

Complete step by step answer:
We are given that ${{x}^{2002}}-2001$ is divided by ${{x}^{91}}$.
We need to find the quotient and the remainder of the above-given division.
So, let us divide ${{x}^{2002}}-2001$ with ${{x}^{91}}$.
$\Rightarrow {{x}^{91}}\overset{{}}{\overline{\left){{{x}^{2002}}-2001}\right.}}$
First let us consider the term ${{x}^{2002}}$ in ${{x}^{2002}}-2001$.
So, we need to multiple ${{x}^{91}}$ with some power of x to divide ${{x}^{2002}}$. Then we get,
$\Rightarrow {{x}^{91}}\overset{{{x}^{1911}}}{\overline{\left){\begin{align}
  & {{x}^{2002}}-2001 \\
 & \underline{{{x}^{2002}}\ \ \ \ \ \ \ \ \ \ \ } \\
 & \ \ \ \ \ \ \ -2001 \\
\end{align}}\right.}}$
So, we get the quotient as ${{x}^{1911}}$ and the remainder as -2001.
As we see the answers in the options have the quotient as the powers of x as factors of 91.
So, the power of x in our quotient is 1911. Then we can write it as,
$\Rightarrow 1911=91\times 21$
So, we can write ${{x}^{1911}}$ as,
$\Rightarrow {{x}^{1911}}={{x}^{91\times 21}}$
So, we get that quotient and the remainder are ${{x}^{91\times 21}}$ and -2001 respectively.
Hence answer is Option C.

Note:
We can also solve this question in an alternate way.
We are given that ${{x}^{2002}}-2001$ is divided by ${{x}^{91}}$.
We need to find the quotient and the remainder for the above-given division.
As we see power of x in ${{x}^{2002}}$ is greater than power of x in ${{x}^{91}}$, ${{x}^{2002}}$ is divisible by ${{x}^{91}}$. So, remainder is -2001.
As ${{x}^{2002}}$ is divisible by ${{x}^{91}}$, the quotient is,
$\begin{align}
  & \Rightarrow {{x}^{2002-91}} \\
 & \Rightarrow {{x}^{1911}} \\
 & \Rightarrow {{x}^{91\times 21}} \\
\end{align}$
So, we get that quotient and the remainder are ${{x}^{91\times 21}}$ and -2001 respectively.
Hence answer is Option C.