Prove that for all numbers abc if a divides bc then aab divi

Prove that for all numbers a,b,c, if a divides bc, then a/(a,b) divides c.

Solution

For a to divide bc it has to have common factors with b and c. Consider a case where a has common factors with b but does not divide it:

a = 10
b = 6

They have a common factor of 2. The question is then whether any c can exist so that a can divide bc without dividing c itself. The answer is yes: Any number that has the remaining factor of 5 can be used. c=5 would work just fine.

a = 10

b = 6

c = 5

a can divide neither number on their own, but it can divide their product just fine.