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 symbol : denotes division.

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neela | High School Teacher | (Level 3) Valedictorian

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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.

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