Problem - 4155
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 $X$?
Answer
36
If $X$ in written in base $7$, it should be in the form of $\overline{a1}_{(7)}$ because the problem suggests $X\equiv 1\pmod{7}$.
Now, after one day, the remaining number of cookies will be $6a_{(10)}$. Therefore, we have $$6a\equiv 2\pmod{7} \implies a \equiv 5 \pmod{7}$$.
Testing $51_{(7)}=\boxed{36}_{(10)}$ finds it is a solution.