Let a b c be positive integers Prove that if a bc and gcda

Let a, b, c be positive integers. Prove that if a | bc and gcd(a, b) = 1 then a | c.

Solution

We know

that a|bc also

gcd(a,b) = 1;

that means b and a do not have any common factor.

and a divides bc;

which can be possible if and only if a divides c, because it cannot divide b.

so thats why if a | bc and gcd(a, b) = 1 then a | c.