Question
The set is considered
$$M= {\frac{1}{\sqrt{1}+\sqrt{2}}, \frac{1}{\sqrt{2}+\sqrt{3}} ,..., \frac{1}{ \sqrt{2022}+\sqrt{2023}} }$$
Show that there are at least 946 subsets of M for which the sum of their elements is a natural number.
My idea
First of all, I observed that all the numbers in the set have the form $\frac{1}{\sqrt{a}+\sqrt{a+1}}=\sqrt{a+1}-\sqrt{a}$ which will make the set:
$$M={\sqrt{2}-\sqrt{1}, \sqrt{3}-\sqrt{2}}, ..., \sqrt{2023}-\sqrt{2022}$$
between $\sqrt{1}$ and $\sqrt{2023}$ there are 44 perfect squares, which means 44 elements of M are the difference between a natural number and an irrational number.
My idea is that if we make a subset that has any two of these 44 elements and the elements between them the sum of the elements of this subset would be a natural number. (*)
So we can make at least $44*43:2=946$ subsets of M for which the sum of their elements is a natural number.
Can you tell me if my solution is ok? If yes, can you help me explain better (*) part?
Hope one of you can help me! Thank you!