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= ma
where , 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, 9
So, (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.