Problem

Source: Iran MO 3rd round 2023 , NT exam , P3

Tags: number theory



Let $K$ be an odd number st $S_2{(K)} = 2$ and let $ab=K$ where $a,b$ are positive integers. Show that if $a,b>1$ and $l,m >2$ are positive integers st:$S_2{(a)} < l$ and $S_2{(b)} < m$ then : $$K \leq 2^{lm-6} +1$$($S_2{(n)}$ is the sum of digits of $n$ written in base 2)