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An integer \(y\) is called the inverse of \(x\) modulo 5 if \(xy \equiv 1 \pmod{5}\).
  1. Determine a modular inverse of \(x = 2\) modulo 5.
  2. Determine a modular inverse modulo 5 for \(x = 3\) and \(x = 4\).
  3. Does \(x = 5\) admit an inverse modulo 5? Why?
  4. Using a congruence table for \(x \in \{0, 1, 2, 3, 4\}\), determine the modular inverse for each value of \(x\).
  5. Use the table to solve the following equations:
    1. \(2x \equiv 3 \pmod{5}\)
    2. \(9x \equiv 1 \pmod{5}\)

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