## Task 1 - 150711

Two mathematicians are discussing their different phone numbers. It turned out the following:

(1) Each of the two phone numbers is a three-digit prime number.

(2) Each individual digit in the two phone numbers, when considered as a single-digit number, is also a prime number.

(3) The digits in the tens place of the two phone numbers are the same. The digit in the hundreds place of one phone number is the digit in the units place of the other, and vice versa.

Determine the phone numbers, and justify the result without using a table of prime numbers as evidence!
The single-digit prime numbers are 2, 3, 5, 7.

Due to (1), the numbers cannot be divisible by 2 or 5, so according to (2), they must end in 3 or 7. Since the numbers are different and due to (3) and (1), the numbers have the form $3 x 7$ or $7 x 3$, where $x$ is a single-digit prime number, i.e., one of the numbers 2, 3, 5, 7.

It is $x \neq 2$; because $3 \mid 327$.

It is $x \neq 5$; because $3 \mid 357$.

It is $x \neq 7$; because $13 \mid 377$.

For $x=3$, the numbers obtained are 337 and 737. Since the first is not divisible by 2 or by $3, 5, 7, 11, 13, 17$ and $19^2=361>337$ is excluded, 337 is a prime number. Similarly, 733 is not divisible by 2 or by $3, 5, 7, 11, 13, 17, 19, 23$ and since $29^2=841>733$, it is a prime number.

Thus, the two mathematicians have the phone numbers 337 and 733.