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= 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.