Question

Show that If d|ab and gcd(a,b)=1 then d|b.

Answer 1

I think you mean gcd(a,d) = 1 right?

Answer 2

oh yeah. sorry.

Answer 3

a can be divided into d evenly, and so does b.
We have evidence of that by looking at that if gcd(a,d) = 1, b has to go into d.

Answer 4

If d|ab, then (ab)/d is an integer by definition


But since gcd(a,d) = 1, this means that d and a have no common factors. So d/a cannot be reduced any further. Assuming that a does not equal 1, this would mean that d/a is NOT an integer.


But (ab)/d is an integer, so factors in d MUST cancel out with factors in b in order for d to effectively go away. Because none of the factors in d cancel with factors in a, this means that ALL of the factors in d cancel with some of the factors in b.


So this shows us that b/d is an integer


This then means that d | b