Question

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

Answer 1

Hint:
Simplify the given expression by rewriting $2002 = 91 \times 22$ in the power of \[x\] . Then, write the dividend as the sum of remainder and product of divisor and quotient. And, if a number $a$ is expressed as $a = bq + r$, then $b$ is the quotient and $r$ is the remainder. Determine the quotient and remainder from the statement obtained.

Complete step by step solution:
We will first simplify the expression.
Since we have to divide ${x^{2002}} - 2001$ by ${x^{91}}$, we will expression ${x^{2002}}$in terms of ${x^{91}}$
As we know, $2002 = 91 \times 22$
We can write ${x^{2002}} - 2001$ as ${x^{91 \times 22}} - 2001$
Therefore, we have to find, $\dfrac{{{x^{91 \times 22}} - 2001}}{{{x^{91}}}}$ which is equivalent to ,
$
  \dfrac{{{x^{91 \times 22}} - 2001}}{{{x^{91}}}} \\
  {x^{91 \times 22}} - 2001 = {x^{91}}\left( {{x^{91 \times 21}}} \right) - 2001 \\
$
If a number $a$ is expressed as $a = bq + r$ , then $b$ is the quotient and $r$ is the remainder.
Therefore, we can say that, the quotient of ${x^{2002}} - 2001$ when it is divided by ${x^{91}}$ is ${x^{91 \times 21}}$ and the remainder of ${x^{2002}} - 2001$ when it is divided by ${x^{91}}$ is $ - 2001$

Hence, option C is correct.

Note:
In these types of questions, we must attempt to write the dividend as the sum of remainder and product of divisor and quotient. It can be done by simplifying the expression using the formulas of exponents such as, ${a^m}{a^n} = {a^{m + n}}$ , $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$ and ${a^{mn}} = {a^m} \times {a^n}$.