<s>[INST] Maximize the given variable under the following constraint Find the greatest integer $a$ such that $a^5 + 5^5$ is divisible by $a - 5$
 [/INST] Make a polynomial (euclidean) division of $a^5+5^5$ by $a-5$ : since
$$
a^5+5^5=(a^5-5^5)+2(5^5)=(a-5)(a^4 + 5a^3 + 25a^2 + 125a + 625)+2(5^5)
$$
If $a$ is a solution, then $a-5$ must divide $2(5^5)$. It is then easy to see
that the largest solution is $a=2(5^5)+5=6255$.
</s>