Let $k(a)$ denote the number of points $(x, y)$ in the plane coordinate system for which $1 \leqq x \leqq a$ and $1 \leqq y \leqq a$ are relatively prime integers. Determine the following sum:

$$
\sum_{i=1}^{100} k\left(\frac{100}{i}\right)
$$
Instead of the questioned sum, we will determine the more general sum $\sum_{i=1}^{n} k(n / i)$.

By definition, $k(n / i)$ gives the number of points $(x, y)$ such that $1 \leqq x \leqq n / i, 1 \leqq y \leqq n / i$, and $x, y$ are coprime, which can also be viewed as the number of points $\left(x^{\prime}, y^{\prime}\right)$ such that $1 \leqq x^{\prime} \leqq n, 1 \leqq y^{\prime} \leqq n$, and $x^{\prime}, y^{\prime}$ are integers whose greatest common divisor is $i$, by introducing $x^{\prime}=i x$ and $y^{\prime}=i y$. Thus, in the sum $\sum_{i=1}^{n} k(n / i)$, we count the points $\left(x^{\prime}, y^{\prime}\right)$ whose both coordinates are integers between 1 and $n$, specifically at the $i$ which is the greatest common divisor of $x^{\prime}$ and $y^{\prime}$.

On the other hand, every such point is counted only once, since the greatest common divisor of two positive integers is uniquely determined. The number of these points is clearly $n^{2}$, hence

$$
\sum_{i=1}^{n} k(n / i)=n^{2}
$$

Lukács 258 ERzsébet (Budapest, Fazekas M. Gyak. Gimn., Iv. o. t.)