Q85 / 160 · WBJEE 2011 · Mathematics

The number \((101)^{100}-1\) is divisible by

  • Correct answer
    A
    \(10^4\)
  • B
    \(10^6\)
  • C
    \(10^8\)
  • D
    \(10^{12}\)

Solution

\(\begin{aligned}\text {Hints: } & (101)^{100}-1={ }^{100} \mathrm{C}_1 100+{ }^{100} \mathrm{C}_2 100^2+{ }^{100} \mathrm{C}_3 100^3+\ldots \ldots +{ }^{100} \mathrm{C}_{100} 100^{100} \\ & =100^2\left[1+{ }^{100} \mathrm{C}_2+{ }^{100} \mathrm{C}_3 100+\ldots \ldots .\right] \\ & =\left(10^4\right) \end{aligned}\)

Hence, the correct answer is option: 'A'.

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