Pam chose some numbers from $1$ to $300$ and wrote them down. As she observed her list, she noticed a peculiar fact that no two numbers on this list added up to a multiple of $3$. What can be the maximum possible number of numbers on this list?
There are exactly $200$ numbers $($ignoring the set $\{ 3,6,9 ,\cdots, 300 \} )$ between $1$ and $300$ which are not divisible by $3$; now among these we have to choose all such that $ \forall a,b$ in the list such that if $a \equiv x_1 \pmod 3$, $\space b \equiv x_2 \pmod 3$ then, $x_1+x_2 \not\equiv 0 \pmod 3 $. Which is the fastest method for counting all such unique $a$ and $b$?