Prove: Forall a, b, c elementof Z, if a does not divide bc, then a does not divide b.

Answer with explanation:It is given that , a,b and c are three elements of Z.Where Z is set of Integers.It is also given that, a does not divide bc.⇒We will use following theorem to prove this.If a divides b, it means¬ b= mawhere , m is an integer.--As, a,  does not divide bc, then, b c will not be integral multiple of a.That is, b c≠ k a→Suppose factor of bc are=1, s, s h, s²h,s²h²,s³h,......,b,........c.Neither of the factors of bc will be divisible by a.→It means (bc ,a) are coprime.For example (7,9) are coprime.Factors of 9 are =1, 3, 9So, (7,3) will be also coprime.→So, all the factors of bc ,which is equal to ={1, s, s h,s²h,s²h²,s³h,......,b,........c} will be coprime with a.⇒So, a and b will be coprime as well as a and c will be coprime.which proves that, a does not divide b.