TITLE: Prove that $f(x)=8$ for all natural numbers $x\ge{8}$
QUESTION [10 upvotes]: A function $f$ is such that $$f(a+b)=f(ab)$$ for all natural numbers $a,b\ge{4}$ and $f(8)=8$. Prove that $f(x)=8$ for all natural numbers $x\ge{8}$

REPLY [2 votes]: For $x\geq 4$, we have $$f(x+5)=f(5x)=f(4x+x)=f(4x^2)=f(2x\cdot 2x)=f(4x)=f(x+4)$$ so $f$ is constant over $[8,\infty)$ as desired.