Do there exist positive integers $a$ and $b$ such that each of the numbers $2^a+3^b,$ $3^a+5^b$ and $5^a+2^b$ is divisible by 29?
Problem
Source: Kyiv mathematical festival 2018
Tags: Kyiv mathematical festival, number theory, Divisibility
Source: Kyiv mathematical festival 2018
Tags: Kyiv mathematical festival, number theory, Divisibility
Do there exist positive integers $a$ and $b$ such that each of the numbers $2^a+3^b,$ $3^a+5^b$ and $5^a+2^b$ is divisible by 29?