A secret agent is trying to decipher an access code. So far, he has obtained the following information:

- it is a four-digit number,
- it is not divisible by seven,
- the digit in the tens place is the sum of the digit in the units place and the digit in the hundreds place,
- the number formed by the first two digits of the code (in this order) is fifteen times the last digit of the code,
- the first and last digits of the code (in this order) form a prime number.

Is this information enough for him to decipher the code? Justify your conclusion.

(M. Petrová)
We will use the given information in such an order that it helps us the most at any given moment.

According to the first piece of information, we are looking for a four-digit number, and according to the fourth piece of information, we will list the possibilities that come into question:

$$
\begin{aligned}
& 15 * 1 \\
& 30 * 2 \\
& 45 * 3 \\
& 60 * 4 \\
& 75 * 5 \\
& 90 * 6
\end{aligned}
$$

(Other possibilities are not valid because $15 \cdot 7=105>99$.)

According to the third piece of information, we will fill in the missing digit:

1561

3022

4583

6044

$75 * 5$ cannot be completed

9066

According to the second piece of information, we are looking for numbers that are not divisible by seven:

1561 does not meet the criteria

$3022 \quad$ meets the criteria

4583 meets the criteria

$6044 \quad$ meets the criteria

$9066 \quad$ meets the criteria

According to the fifth piece of information, we will create the given two-digit numbers and select the options where we get a prime number:

3022 creates 32, does not meet the criteria

4583 creates 43, meets the criteria

6044 creates 64, does not meet the criteria

9066 creates 96, does not meet the criteria

Of all the four-digit numbers, only one satisfies all five conditions, and that is 4583. This means that this information is sufficient for the secret agent to decipher the access code.

Note. If we use the fifth piece of information before the second, we will exclude all even numbers (the created two-digit number will be even, so it will not be a prime number). For checking divisibility by seven, only the numbers 1561 and 4583 remain.

Evaluation. 2 points for identifying the numbers using the fourth piece of information; 3 points for progressively eliminating numbers using additional information; 1 point for explaining that this information is sufficient for the agent.