Question 56
For each positive integer $n$, let $\operatorname{rad}(n)$ denote the product of the distinct prime factors of $n$. Show that there exist integers $a, b \gt 1$ such that $\operatorname{gcd}(a, b)=1$ and
$$
\operatorname{rad}(a b(a+b)) \lt \frac{a+b}{2024^{2024}}
$$
For example, $\operatorname{rad}(20)=\operatorname{rad}\left(2^{2} \cdot 5\right)=2 \cdot 5=10$ and $\operatorname{rad}(18)=\operatorname{rad}\left(2 \cdot 3^{2}\right)=2 \cdot 3=6$.
$$
\operatorname{rad}(a b(a+b)) \lt \frac{a+b}{2024^{2024}}
$$
For example, $\operatorname{rad}(20)=\operatorname{rad}\left(2^{2} \cdot 5\right)=2 \cdot 5=10$ and $\operatorname{rad}(18)=\operatorname{rad}\left(2 \cdot 3^{2}\right)=2 \cdot 3=6$.