7.1. In the thirtieth kingdom, there are only two types of coins: 16 and 27 tugriks. Can you pay for a notebook that costs 1 tugrik and get change?
Answer: Yes, it is possible.

Solution. For example, you can pay with three coins of 27 tugriks and receive change with five coins of 16 tugriks.

Other examples can be provided in a general form:

a) pay with $3+16$ coins of 27 tugriks and receive change with $5+27 n$ coins of 16 tugriks, where $n$ is a natural number;

b) pay with $22+27$ coins of 16 tugriks and receive change with $13+16$ m coins of 27 tugriks, where $m$ is a natural number or zero.

These examples are derived from the following considerations. It is clear that paying and receiving change with coins of the same denomination is meaningless. Therefore, it is necessary to find such integers $x$ and $y$ that satisfy the equation $16 x+27 y=1$.

Note that an equation of the form $a x+b y=c$ has a solution in integers if and only if the number $c$ is divisible by the GCD $(a ; b)$.

## Grading Criteria:

+ correct answer and example provided

$\pm$ several examples provided, some of which are correct and some are incorrect

- problem not solved or solved incorrectly