Question 124
There are five piles of $42$, $70$, $105$, $462$, and $2009$ stones respectively. You are allowed to do the following: instead of two piles consisting of $a$ and $b$ stones you can make
* two piles of $a$ and $(a+b)$ stones;
* two piles of $a$ and $|a-b|$ stones.
You can also choose $k$ different piles in which the total number of stones equals $nk$, where $n$ is some natural number, and make
* $k$ piles of $n$ stones each instead.
Can we make five piles each of which consists of
a) $2008$ stones;
b) $2009$ stones
in a finite number of steps?
* two piles of $a$ and $(a+b)$ stones;
* two piles of $a$ and $|a-b|$ stones.
You can also choose $k$ different piles in which the total number of stones equals $nk$, where $n$ is some natural number, and make
* $k$ piles of $n$ stones each instead.
Can we make five piles each of which consists of
a) $2008$ stones;
b) $2009$ stones
in a finite number of steps?