Question

Let a, b in Z. Denote by D(a) the set of all divisors of a, and by D(b)
the set of all divisors of b. Prove that a|b if and only if D(a) is in subset of D(b).

Answer 1

sorry having trouble writing proper symbols here

Answer 2

First, let a|b. Let d be a divisor of a. If d|a, and a|b, then by transitivity, d|b. Therefore all the divisors of a also divide b, and D(a) is a subset of D(b).

Conversely, if D(a) is a subset of D(b), then every divisor of a is a divisor of b. that gives us:
\[d\mid a\Longrightarrow d\mid b\] for every divisor of a. Thus a must divide b, because if a didnt, there would have to be at least one divisor of a that didnt divide b, which contradicts the assumption that all the D(a) is a subset of D(b)

Answer 3

thanks joe, can i just ask what does the symbol | actually show? or mean?