Summary of Techniques for Sequence Limits

Denote by $\langle{x}\rangle$ the fractional part of the real number $x$ (for instance, $\langle{3.2}\rangle = 0.2$). A positive integer $N$ is selected randomly from the set $\{1, 2, 3, \dots , M\}$, with each integer having the same probability of being picked, and $\langle{87N/303}\rangle$ is calculated. This procedure is repeated $M$ times and the average value $A(M)$ is obtained. What is $303\lim_{M\to\infty} A(M)$?

Source: HMMT February 2002