A pairwise balanced design is a set of elements $X$ and set of blocks $A$ such that each pair of elements of $X$ occurs in exactly $\lambda$ blocks.
I am trying to solve the following problem: Given a PBD with $|X|=v$, where $v \equiv 2\mod 3$, $\lambda = 1$ and every block $B \in A$ is of size $k$ or size $3$, then $k\equiv 2\mod 3$.
I'm really not sure where to start. Any help would be really appreciated!
Suppose there are $m$ blocks of size $k$ and $n$ of size $3$, then $3n=\binom{v}{2}-m\binom{k}{2}$, notice $\binom{v}{2}$ is congruent to $1\bmod 3$. Unless $k$ is congruent to $2\bmod 3$ we shall have $\binom{k}{2}$ is a multiple of three. This would mean $\binom{v}{2}-m\binom{k}{2}$ would not be a multiple of $3$, and $n$ would then not be an integer.