The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, where $a > 0$. Find the base-$10$ representation of $n$.

Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.

Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_{b}$ is a perfect square and the base-$b$ integer $27_{b}$ is a perfect cube.

A person eats $X ( > 1)$ cookies in $N$ days in the following way:

• He eats $1$ plus $1/7$ of the remaining cookies on the $1^{st}$ day 
• He eats $2$ plus $1/7$ of the remaining cookies on the $2^{nd}$ day
• $\cdots$
• Finally, he eats the last $N$ cookies on the $N^{th}$ day

What is the smallest possible value of $X$?

Let the binary representation of positive integer $n$ be $b_tb_{t-1}\cdots b_1b_0$. Show that $$\binom{n}{2^j} \equiv b_j \pmod{2}$$

where $j$ is a non-negative integer. Note that $\binom{n}{m} = 0$ if $m > n$.

Let $n$ be a positive integer and $k$ be the number of $1$s in $n$'s binary representation. Show there are $2^k$ odd integers in $\binom{n}{0}$, $\binom{n}{1}$, $\cdots$, $\binom{n}{n}$.