Find counterexamples to each of these statements about congruences.a) If ac ? bc (mod m), where a, b, c, and m are integers with m ? 2, then a ? b (mod m).________________b) If a ? b (mod m) and c ? d (mod m), where a, b, c, d, and m are integers with c and d positive and m ? 2, then ac ? bd (mod m).

Solution:Step 1In this problem we have to find counterexamples to these statements about congruences.we can say integer a is congruent to b (mod) m, if a and b are integers and m is a positive integer and m divides( a-b) completely that is m | (a-b).Or,We can also say a is congruent to b (mod) m. if value of Step 2a) If ac bc (mod m), where a, b, c, and m are integers with m 2, then a b (mod m).For counterexample let, a=4, b=5, c=6, m=6.Here, for ac bc (mod m)We have For ac bc (mod m) to satisfy it should be follow congruence theorem which is true for given equation.now , for a b (mod m).We have, a=4, b=5 and m=6 which does not satisfy congruence theorem that is m | (a-b) for a b (mod m).Hence with counterexample we proved that...