Consider the following statement For all integers a b and c

Consider the following statement: For all integers a, b, and c, if a|b and a|c then a|(2b - 3c). Prove the statement. Write the contrapositive of the statement. Write the converse of the statement and determine if it is true or false. Justify your answer.

Solution

Given a/b and a/c

from the definition of divisible b=aq and c=ak where q and k are integers.

consider 2b-3c = 2(aq)-3(ak) = a(2q-3k) = am

where m=2q-3k is also an integer.

Therefore by the definition of divisible we have that a divides 2b-3c evenly.