## Task A-4.4.

Tamara wrote an even natural number on the board. Then, one by one, she wrote twelve more numbers such that each number was 5 greater than the square of the previously written number. Determine which digit the last written number can end with.
## First solution.

If the number written at the beginning ends in 0, then in the next step it will end in 5 $\left(0^{2}+5=5\right)$, and then again with 0 $\left(5^{2}+5=30\right)$, and this will repeat periodically. After 12 steps, the number will end in 0.

If the number written at the beginning ends in 4 or 6, then in the next step it will end in 1 $\left(4^{2}+5=21,6^{2}+5=41\right)$, and then again with 6 $\left(1^{2}+5=6\right)$, and this will also repeat periodically. After 12 steps, the number will end in 6.

If the number written at the beginning ends in 2 or 8, then in the next step it will end in 9 $\left(2^{2}+5=9,8^{2}+5=69\right)$, then with 6 $\left(9^{2}+5=86\right)$, then again with 1, 6, etc. After 12 steps, we are left with a number that ends in 6.

Therefore, at the end, the number on the board can end with the digit 0 or 6.