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CMI2021-B: 1

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Let $\text{A}=\left ( \sqrt{3} +\sqrt{2}\right )^{1000}, \text{B}=\left ( \sqrt{3} -\sqrt{2}\right )^{1000}$. Prove the following statements.

  1. $\text{A+B}$ is an integer.

  2. The digit immediately after the decimal point in $\text{B}$ is $0$.

  3. The digit immediately after the decimal point in $\text{A}$ is $9$.

  4. Both $\text{A, B}$ is irrational.

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