34. Find the number of pairs of positive integers $(x, y)$ are there which satisfy the equation $2 x+3 y=2007$.
34. Answer: 334
$$
2 x+3 y=2007 \Rightarrow 2 x=2007-3 y=3(669-y) \text {. }
$$

Since 2 and 3 are relatively prime, it follows that $x$ is divisible by 3 . Write $x=3 t$, where $t$ is a positive integer.
The equation reduces to $y=669-2 t$.
Since $669-2 t>0$, we have $t<334.5$, and since $t$ is a positive integer, we have $1 \leq t \leq 334$.
Conversely, for any positive integer $t$ satisfying $1 \leq t \leq 334$, it is easily seen that $(3 t, 669-2 t)$ is a pair of positive integers which satisfy the given equation.
Therefore there are 334 pairs of positive integers satisfying the given equations.