The number $12$ is written on the whiteboard. Each minute, the number on the board is either multiplied or divided by one of the numbers $2$ or $3$ (a division is possible only if the result is an integer) . Prove that the number that will be written on the board in exactly one hour will not be equal to $54$.
Let $x_n$ be the number that is written on the whiteboard after exactly $n$ minutes have passed. We know that $x_0=12.$ Also, if $x_i = 2^a \cdot 3^b,$ for some nonnegative integers $a$ and $b,$ let $a_i=a+b.$ Since $12=2^2\cdot3,$ we know that $a_0=3.$
Clearly, every minute, the $a_i$ value will increase or decrease by $1,$ so it follows that the parity remains invariant after an even number of minutes. However, note that $54=2 \cdot 3^3.$ Since $1+3=4,$ which is even, this is a contradiction since the parity of $4$ is not the same as the parity of $3.$
Hence, the number written on the board in exactly one hour cannot be equal to $54.$ $\square$