2. $[\mathbf{1 5}]$ Let $a_{1}, a_{2}, \ldots$ be an infinite sequence of integers such that $a_{i}$ divides $a_{i+1}$ for all $i \geq 1$, and let $b_{i}$ be the remainder when $a_{i}$ is divided by 210 . What is the maximal number of distinct terms in the sequence $b_{1}, b_{2}, \ldots$ ?
Answer: $\quad 127$ It is clear that the sequence $\left\{a_{i}\right\}$ will be a concatenation of sequences of the form $\left\{v_{i}\right\}_{i=1}^{N_{0}},\left\{w_{i} \cdot p_{1}\right\}_{i=1}^{N_{1}},\left\{x_{i} \cdot p_{1} p_{2}\right\}_{i=1}^{N_{2}},\left\{y_{i} \cdot p_{1} p_{2} p_{3}\right\}_{i=1}^{N_{3}}$, and $\left\{z_{i} \cdot p_{1} p_{2} p_{3} p_{4}\right\}_{i=1}^{N_{4}}$, for some permutation $\left(p_{1}, p_{2}, p_{3}, p_{4}\right)$ of $(2,3,5,7)$ and some sequences of integers $\left\{v_{i}\right\} \cdot\left\{w_{i}\right\} \cdot\left\{x_{i}\right\} \cdot\left\{y_{i}\right\} \cdot\left\{z_{i}\right\}$, each coprime with 210.

In $\left\{v_{i}\right\}_{i=1}^{N_{0}}$, there are a maximum of $\phi(210)$ distinct terms mod 210. In $\left\{w_{i} \cdot p_{1}\right\}_{i=1}^{N_{1}}$, there are a maximum of $\phi\left(\frac{210}{p_{1}}\right)$ distinct terms $\bmod 210$. In $\left\{x_{i} \cdot p_{1} p_{2}\right\}_{i=1}^{N_{2}}$, there are a maximum of $\phi\left(\frac{210}{p_{1} p_{2}}\right)$ distinct terms mod 210. In $\left\{y_{i} \cdot p_{1} p_{2} p_{3}\right\}_{i=1}^{N_{3}}$, there are a maximum of $\phi\left(\frac{210}{p_{1} p_{2} p_{3}}\right)$ distinct terms mod 210. In $\left\{z_{i} \cdot p_{1} p_{2} p_{3} p_{4}\right\}_{i=1}^{N_{4}}$, there can only be one distinct term $\bmod 210$.
Therefore we wish to maximize $\phi(210)+\phi\left(\frac{210}{p_{1}}\right)+\phi\left(\frac{210}{p_{1} p_{2}}\right)+\phi\left(\frac{210}{p_{1} p_{2} p_{3}}\right)+1$ over all permutations $\left(p_{1}, p_{2}, p_{3}, p_{4}\right)$ of $(2,3,5,7)$. It's easy to see that the maximum occurs when we take $p_{1}=2, p_{2}=$ $3, p_{3}=5, p_{4}=7$ for an answer of $\phi(210)+\phi(105)+\phi(35)+\phi(7)+1=127$. This upper bound is clearly attainable by having the $v_{i}$ 's cycle through the $\phi(210)$ integers less than 210 coprime to 210 , the $w_{i}$ 's cycle through the $\phi\left(\frac{210}{p_{1}}\right)$ integers less than $\frac{210}{p_{1}}$ coprime to $\frac{210}{p_{1}}$, etc.