Q:

Application of invariant method

On the board is written the number $18$. Every minute the number is replaced by the product with 2 or 3,or by the quotient of the division with 2 or 3.  Show that after $60$ minutes the number cannot be $96$.  
What I tried is to factorize into prime: $18=2\cdot3^2$ and $96=2^5\cdot3$, but I don't see what the invariant is, since the division by $2$ or $3$ may result in new prime factors. 

A:

Hint: At any time, the number has the shape $2^a\cdot 3^b$. A useful invariant is the parity of $a+b$, that is, the remainder when $a+b$ is divided by $2$.
The parity changes every minute, so it must be unchanged after an even number of minutes.