573. Do there exist natural numbers $a, b$, and $c$ such that the decimal representation of the sum

$$
a^{60}+b^{60}+c^{60}
$$

ends in the digit 9?

574. Prove that the square of a two-digit number $\overline{a b}(b \neq 0)$ ends in $\overline{a b}$ only when $\overline{a b}$ is 25 or 76.
$\triangle$ Numbers $x=\overline{a b}$ and $x^{2}=\overline{a b^{2}}$ end with the same two digits if and only if their difference is divisible by 100:

$$
\left(x^{2}-x\right): 100, \quad x(x-1) \vdots 100
$$

Two consecutive natural numbers $x-1$ and $x$ are coprime (see the supporting problem 454 from § 11), so their product is divisible by 100 only when one of them is divisible by 25 and the other by 4. (It is necessary to consider that these factors are two-digit numbers, so none of them can be divisible by 100.)

Consider two cases.

1) Let the number $x$ be divisible by 25, and $x-1$ by 4.

From the first condition, we find that $x \in\{25 ; 50 ; 75\}$. Of these numbers, only 25 satisfies the second condition.

2) Let $x$ be divisible by 4, and $x-1$ by 25.

From the second condition, we get that $(x-1) \in\{25 ; 50 ; 75\}$, i.e., $x \in\{26 ; 51 ; 76\}$. The first condition is satisfied only by 76.