Problem

Source: Indonesian National Mathematical Olympiad 2024, Problem 2

Tags: number theory, Inamo, primes, factorizations, Indonesia, NT construction, Indonesia MO



The triplet of positive integers $(a,b,c)$ with $a<b<c$ is called a fatal triplet if there exist three nonzero integers $p,q,r$ which satisfy the equation $a^p b^q c^r = 1$. As an example, $(2,3,12)$ is a fatal triplet since $2^2 \cdot 3^1 \cdot (12)^{-1} = 1$. The positive integer $N$ is called fatal if there exists a fatal triplet $(a,b,c)$ satisfying $N=a+b+c$. (a) Prove that 16 is not fatal. (b) Prove that all integers bigger than 16 which are not an integer multiple of 6 are fatal.