Q66 / 155 · WBJEE 2018 · Mathematics

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

  • Correct answer
    A
    $10^{4}$
  • B
    $10^{6}$
  • C
    $10^{8}$
  • D
    $10^{12}$

Solution

$(101)^{100}-1=(1+100)^{100}-1$ $=\left(1+{ }^{100} C_{1} \cdot 100+{ }^{100} C_{2} 100^{2}+\ldots \ldots\right)-1$ $={ }^{100} C_{1} 100+{ }^{100} C_{2}(100)^{2}+$ $\quad{ }^{100} C_{3}(100)^{3}+\ldots \ldots+{ }^{100} C_{100}(100)^{100}$ $=10^{4}\left(1+{ }^{100} C_{2}+{ }^{100} C_{3} 10^{2}+\ldots .\right.$ $\quad+{ }^{100} C_{100}(100)^{98}$ $=10^{4}(1+$ an integer multiple of 10$)$

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

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