Solution:

$9$

Step by step solution:

Let $x$ and $y$ be the number of apples in each packet and the number of apples of each people, respectively

We get the equation as following:

$\quad 37x - 17 = 79y$

Or: $37x - 79y = 17\qquad (*)$

Because $x$ and $y$ are integers

So $(*)$ is equation with integer solutions

We have $(x;y)= (9;4)$ as an particular solution of $(*)$

Therefore, the formula of all solutions of $(*)$ is:

$\begin{cases}x = 9 + 79t\\y = 4 + 37t\end{cases}\quad (t\in\Bbb Z)$

Beside, $x$ is the smallest positive integer

Or: $x \geqslant 1$

So that: $9 + 79t \geqslant 1$

It's equal to: $t \geqslant -\dfrac{8}{79}$

We can rewrite $t$ as: $t\in \{0;1;2;\dots\}$

Choose the smallest $t$ to get the smallest $x$, we get: $t = 0$

Therefore: $x = 9+ 79.0 = 9$

Answer: $9$

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