Question 122
Find the smallest real number $\lambda$ with the following property: For any positive integers $a$, $b$, $n$ where $a + b$ is not divisible by $n$, there exists a positive integer $k \lt n$ such that
$$
\left\{ \frac{ak}{n} \right\} + \left\{ \frac{bk}{n} \right\} \le \lambda,
$$
where $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.
$$
\left\{ \frac{ak}{n} \right\} + \left\{ \frac{bk}{n} \right\} \le \lambda,
$$
where $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.