number theory questions

Let's look at the scenario when n = 1, the only possible solution is both x & y are 2. Actually, it is the same for other number. For a integer n, when the number is split into two half, there are two possibilities. One is both half are the same, i.e. x = y = 2n. Another possibility is one half is larger than the other half. When we represent the smaller half as 1/x, you will find that it is impossible to find a integer y that can represent a larger portion. To represent it mathematics formula, you may see below

Assume there is available x and y that fulfill the formula:

1/x + 1/y = 1/n

In order to make fulfill this equation, both x and y must be greater than n,
and both x and y must be multiple of n.

so assume x= k*n, y = m*n, (both m and k are integers), the formula can be rewrite as:

1/kn + 1/mn = 1/n, or
1/k + 1/m = 1

since if both k and m are not equal to 2, both 1/k and 1/m will be smaller than 1/2, so there will not be possible to make the formula valid. In another word, for another integer n, the only solution for x and y are 2n.