Question 161
Let $N$ be a natural number and $x_{1}, x_{2}, \ldots, x_{n}$ other natural numbers smaller than $N$ and such that the least common multiple of any two of these $n$ numbers is larger than $N$.
Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$; so
$$
\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}\lt 2
$$
Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$; so
$$
\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}\lt 2
$$