This is not homework. If n is a positive odd integer then n and [itex] n+2^k [/itex] are relatively prime. k is a positive integer. Lets assume for contradiction that n and [itex] n+2^k [/itex] have a common factor. then it should divide their difference but their difference is [itex] 2^k [/itex] and since n is odd it has no factors of 2 so this is a contradiction and they are relatively prime.