Let
be the number of all non-negative integers
satisfying the following conditions:
(1) The integer
has exactly
digits in the decimal representation (where the first digit is not necessarily non-zero!), i. e. we have
.
(2) These
digits of n can be permuted in such a way that the resulting number is divisible by 11.
Show that for any positive integer number
we have
.
![f(k)](/media/m/4/2/4/4240ef816e3c37cc1176590bec5ddc22.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
(1) The integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![0 \leq n <10^k](/media/m/8/5/3/8533c3fe1a4a98150a69b39ef25be529.png)
(2) These
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
Show that for any positive integer number
![m,](/media/m/d/4/8/d4801e2df194f69b71c59917ecff5fcf.png)
![f\left(2m\right) = 10 f\left(2m - 1\right)](/media/m/4/8/0/4804aad0a2eb10f759381550a5ea0007.png)