Task A-4.6. (10 points)

Let $a$ and $b$ be natural numbers. Which digits can be in the units place of the decimal representation of the number $(a+b)^{5}-\left(a^{5}+b^{5}\right)$?
## Solution.

The given expression can be expanded as follows:

$$
\begin{aligned}
& (a+b)^{5}-\left(a^{5}+b^{5}\right) \\
& =a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}-a^{5}-b^{5} \\
& =5\left(a^{4} b+2 a^{3} b^{2}+2 a^{2} b^{3}+a b^{4}\right) \\
& =5 a b\left(a^{3}+2 a^{2} b+2 a b^{2}+b^{3}\right) \\
& =5 a b(a+b)\left(a^{2}+a b+b^{2}\right) .
\end{aligned}
$$

From $(*)$ we see that the given expression is always divisible by 5.

(4 points)

Let's prove that it is also divisible by 2.

This is obviously true if at least one of the numbers $a$ and $b$ is even.

Otherwise, $a$ and $b$ are odd, so $a+b$ is an even number.

Thus, in this case, the given expression is also divisible by 2.

We conclude that the given number ends in the digit 0 for all natural numbers $a$ and $b$.

(1 point)