Independence, part 2

\[ \] A fair die is thrown two times independently. Let \(X\) and \(Y\) be the random variable indicating the outcomes of these two throws. Define \(Z = X + Y,\) and define \(U\) as the remainder when \(Z\) is divided by \(6\).

Which of the following statements are true?

  1. \(X\) and \(Z\) are independent.
  2. \(X\) and \(U\) are independent.
  3. \(Z\) and \(U\) are independent.
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