Problem: Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms. Answer: We see that $\phi(30) = 8$. also, we see that $\phi(30k+30) - \phi(30k) = \phi(30)(\phi(k+1)-\phi(k)) = \phi(30)$ which tells us that between each "block" of $30$ integers, there are exactly $8$ integers $a$ such that $\gcd(a,30) = 1$. Further, we know that $\frac{a}{30} < 100 \implies a < 300$. We find that there are $10$ blocks in total (meaning there are $80$ coprime-with-$30$ integers between $0$ and $300$). The sum of the integers in the first block is $120$. The critical part is that for subsequent blocks, each coprime number increases by multiples of 30 . Specifically, for the $k^{\text {th }}$ block where $k=0$ to $9$, we have that the sum for block $k=120+30 k \times 8=120+240 k$. Thus, the sum over all blocks we find is $12000$ and we divide by $30$ to get $400$.