2. There are two triples $(a, b, c)$ of positive integers that satisfy the equations

$$
\begin{aligned}
& a b+c=34, \\
& a+b c=29 .
\end{aligned}
$$

Which two triples are these?
B2. $(13,2,8)$ and $(5,6,4)$ We subtract the second equation from the first. This gives

$$
a b+c-a-b c=5 .
$$

The left side of this equation can also be written as $(a-c)(b-1)$. We thus find that $(a-c)(b-1)=5$. Since $b$ must be a positive integer, the factor $b-1$ cannot be negative. Since 5 is a prime number, there are then two possibilities: $b-1=1$ and thus $b=2$ or $b-1=5$ and thus $b=6$.

If $b=2$, then we also immediately find that $a-c=5$. If we add this to the equation $a b+c=34$, we get $a+a b=39$. Since $b=2$, this gives $a=13$ and thus $c=8$. If we substitute $(a, b, c)=(13,2,8)$ into the original equations, we see that this is indeed a solution.

If $b=6$, then we also immediately find that $a-c=1$. Now we add this to the equation $a b+c=34$ and find $a+a b=35$. In this case, we had $b=6$ and thus $a=5$. We then also immediately find that $c=4$. If we substitute $(a, b, c)=(5,6,4)$ into the original equations, we see that this is also a solution.

The two solutions are thus $(a, b, c)=(13,2,8)$ and $(a, b, c)=(5,6,4)$.