clearly the order of σ has to be a multiple of 84, since otherwise no element in {1,2....,84} is mapped to itself. what else can you say?
Let σ be the following permutation in the symmetric group S1000. It cyclically permutes all numbers between 1 and 84. This means it sends 1 to 2, 2 to 3, and so on until 84 which is sends back to 1. It also cyclically permutes all numbers between 101 and 317 (sending 101 to 102, 102 to 103, and so on until we get to 317 which σ sends back to 101). And σ leaves all other numbers from 85 to 100 and from 318 to 1000 put. Find the order of σ in S1000.