Are the following statements true or false? Either prove the statement is true or provide a counterexample to show it is false. (a) For all integers a and b, if a . b = 0 (mod 6), then a = 0 (mod 6) or b = 0 (mod 6). (b) For all integers a and b, if a b = 0 b = 0 (mod 8). (c) For all integers a and b, if a b = 1 (mod 6). (mod 8), then a = 0 (mod 8) or b = 1 (mod 6), then a = 1 (mod 6) or

Q: The statement a ≡ b (mod n) is called a congruence.

A: Given that  The statement a ≡ b (mod n) is called a congruence.

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Q: Perform the following calculations. a. (14 + 30) mod 19 b. (14-30) mod 19 C. (3*13) mod 19

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Q: (b) Prove that if a = b (mod m) and if b = c (mod m), then a = c (mod m).

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Q: (c) For each integer a, if a 4 (mod 8), then a 2 (mod 8).

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A: The solution is given as follows

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Q: Proficiency #7. [Counterexamples] For each of the following two statements, give a counterexample to…

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