Question

In: Advanced Math

For each proposition, either give a counterexample showing it is false, or write a proof. (a)...

For each proposition, either give a counterexample showing it is false, or write a proof.
(a) For all a, b, c ∈ Z, if ab divides c then a divides c and b divides c.
(b) For all a, b, c ∈ Z, if a divides bc, then a divides b or a divides c.

Expert Solution

(a) solution:

Given that , i.e. ab divides c, for all

This implies that there exist an integer q such that

and also

and where and are some integers.

Hence, if ab divides c, then a divides b and b divides c.

(b) Given that , i.e. a divides bc, for all .

Suppose . This implies a can not divide b.

Then, [standard result]

Taking , since

,i.e. a divides c.

Similarly, we consider .

Then, , i.e. a divides b.

Hence, if a divides bc, then a divides b or a divides c.

Colby Messinger answered 8 months ago

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