Equivalence classes have the following properties:In define a relation by:

~ if is divisible by .

Show that ~ is an equivalence relation.

1). Reflexivity a~a

2). Symmetry if a~b then b~a

3). Transitivity if a~b and b~c then a~c

1). Reflexivity: so reflexivity holds.

2). Symmetry: if then so symmetry holds.

3). Transitivity: if and then so transitivity holds.

I can see that this is going to have something to do with remainders, but i'm not really sure what. Surely remainders can be repeated?Let A denote the set of equivalence classes of ~ in .

Find an explicit bijection on .

Hint: What if the relation were " ~ if is divisible by "?