By the unique factorization of , write where each is a prime number (distinct from one another) and a positive integer.
If is divisible by , then is divisible by .
If is divisible by , write , where . Since divides , write where is a positive integer. Then . Since , is divisible by . ∎
For example, the subset of all decimal fractions is -divisible. is also and -divisible. In general, we have the following:
If is -divisible, it is also -divisible for every non-negative integer .
Suppose and are coprime, then is -divisible and -divisible iff it is -divisible.
This follows from proposition 1 and the fact that if , and , then . ∎
is -divisible iff is -divisible for every prime dividing .
Remark. is a divisible group iff is -divisible for every prime .
|Date of creation||2013-03-22 17:27:30|
|Last modified on||2013-03-22 17:27:30|
|Last modified by||CWoo (3771)|