Let a,b, and c be integers. Prove or disprove that a:bc implies ac:bcPlease provide detailed steps and explanation. Please note that this...

If a , b , c are any three integer, and if a divides bc, then
implies ac divides bc. To prove or dis[rove the statement.

Since
a|bc , we can find a integer k such that  bc = ak....(1)

Since ac
also divides bc , we canfind an integer n such that bc =
acn...(2)

From (1) and (2) we get acn = ak =
a(bc/a).

acn = a(bc/a).

acn = ac
(b/a).

Therefore if n = b/a is an integer then only if adivides bc,
the ac divides bc.

If b/a is not an integer, then ifa divides bc ,
then ac doenot divide bc.

Example : a = 6, b=12 and c =8 divides bc
= 12*8 =96, ac= 6*8 = 48 divides 96. Here  a divides 12.

Example: a
= 6, b = 8 , c = 12.  bc = 96. 6 divides 96. But 6*12= 72 does not divide 96, as  a= 6 does not
ivide b = 8.