Say that

.
Note that

.
Thus,

.
Our goal now is to show either

. Now

follows at once from the equation

. But if

we will show this implies that

.
Now if

then

. Thus,

.
Now we have established that

.
Say the first one is true

. Then

. But since

. Say therefore that

. Substitute

into

we get

. But then

.
Now using a similar argument with

we find that

.
I hope you like this proof. I really find it beautiful how divisibility arguments lead us to this conclusion.