## Task 1 - 340821

A four-digit natural number is called "symmetric" if and only if its thousands digit is equal to its units digit and its hundreds digit is equal to its tens digit. Tanja claims that every four-digit symmetric number is divisible by 11.

a) Check this divisibility with three self-chosen examples!

b) Prove Tanja's claim in general!

c) How many four-digit symmetric numbers are there in total?

d) How many even four-digit symmetric numbers are there in total?
(a) For example, one confirms $3443: 11=313,5555: 11=505,9009: 11=819$.

(b) Every four-digit symmetric number $z$ can be expressed with two natural numbers $a, b$ as

$$
1000 a+100 b+10 b+a=11 \cdot(91 a+10 b)
$$

thus, since $91 a+10 b$ is a natural number, it is divisible by 11.

(c) In (1), there are 9 possibilities for the digit $a$: $1,2, \ldots, 9$, and independently, 10 possibilities for the digit $b$: $0,1,2, \ldots, 9$. Therefore, there are a total of $9 \cdot 10=90$ four-digit symmetric numbers.

(d) Since $z$ is even if and only if the units digit $a$ is even, there are now 4 possibilities for $a$: 2, $4,6,8$. For $b$, the same possibilities as in (c) apply. Therefore, there are a total of $4 \cdot 10=40$ even four-digit symmetric numbers.