1. There are two types of containers: 27 kg and 65 kg. How many containers of the first and second type were there in total, if the cargo in the first type of containers exceeds the cargo in the second type of containers by 34 kg, and the number of 65 kg containers does not exceed 44 units?
Solution. Let $x$ be the number of containers weighing 27 kg, and $y$ be the number of containers weighing 65 kg. We obtain the equation $27 x - 65 y = 34$.

$27(x - 2y) - 11y = 34$, let $x - 2y = k$.

$27k - 11y = 34$,

$11(2k - y) + 5k = 34$, let $2k - y = t$.

$11t + 5k = 34$,

$5(2t + k) + t = 34$, let $2t + k = n$.

$5n + t = 34, t = 34 - 5n$.

Substitute into (3), $k = 11n - 68$.

Substitute into (2), $y = 27n - 170$.

Substitute into (1), $x = 65n - 408$.

Since $x > 0, 0 < y \leq 44$, then $n = 7$. Therefore, $x = 47, y = 19$ or 47 and 19 containers, a total of 66 containers.

Answer: 66.