Determine which of the following statements are true. Prove each true statement, and give a counterexample for each false statement.
(a) If x = 1 (mod 5), then x = 1 (mod 10).
(b) If x = 0 (mod 6) or x = 3 (mod 6), then x = 0 (mod 3).
(c) If x = 1 (mod 10) or x = 6 (mod 10), then x = 1 (mod 5).
(d) If x = 1 (mod 5), then x = 1 (mod 10) or x = 6 (mod 10).
(e) For all m, n [ N, if x = 0 (mod m) and x = 0 (mod n), then x = 0 (mod mn).
(f) For all m [ N, if x = 0 (mod 2m) or x = m (mod 2m), then x = 0 (mod m).
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