732. A grasshopper jumps along a straight line to the left and right, and it can make large jumps of 36 cm and small jumps of 14 cm (to the left or to the right). Can it move 3 cm from its initial position in several jumps? 2 cm? 1234 cm?
$\Delta$ He cannot move 3 cm, as all his jumps shift him by an even number of centimeters (and by adding or subtracting even shifts, we cannot get an odd number: any shift will be a multiple of 2 cm).

He can move 2 cm. Indeed, the grasshopper can move 22 cm if he jumps 36 cm in one direction and then returns 14 cm in the opposite direction ( $36-14=22$ ). Knowing how to move 22 cm (let's say in two jumps instead of one) and 14 cm, he can move 8 cm $(22-14=8)$. Knowing how to move 14 and 8 cm, he can move 6 cm $(14-8=6)$. Finally, knowing how to move 8 and 6 cm, he can move 2 cm $(8-6=2)$.

Then, by repeating the 2 cm shift multiple times, he can move any even number of centimeters, including 1234. (Since $1234=2 \cdot 617$, he needs to repeat the 2 cm shift 617 times.) $\triangleleft$

Programmers would say this: having commands to shift 36 and 14 cm, we can write a subroutine (procedure, auxiliary algorithm) to shift 22 cm. Then, using this subroutine and the command to shift 14 cm, we write a subroutine to shift $8=22-14$ cm, and so on, until we reach a shift of $2 \mathrm{~cm}$. Now, by repeating the 2 cm shift command 617 times, we can move 1234 cm.

733 (Continuation). How many and what kind of jumps must the grasshopper make to move 2 cm, as in the previous problem?

$\Delta$ This can be calculated by following its solution:

$$
\begin{aligned}
22 & =36-14 \\
8 & =22-14=(36-14)-14=36-2 \cdot 14 \\
6 & =14-8=14-(36-2 \cdot 14)=3 \cdot 14-36 \\
2 & =8-6=(36-2 \cdot 14)-(3 \cdot 14-36)= \\
& =2 \cdot 36-5 \cdot 14
\end{aligned}
$$

This results in 2 jumps in one direction of 36 cm (a total of 72 cm) and 5 jumps in the opposite direction of 14 cm (a total of 70 cm). It all adds up: $72-70=2 . \triangleleft$