Question 404050
We want the least term that is a multiple of the LCM of 2, 3, ..., 6, or 60.


Our sequence is denoted by 17 + 7k, where k is an integer. Writing everything modulo 2, 3, and 5, we obtain


1 + k ≡ 0 (mod 2) --> k ≡ 1 (mod 2)
2 + k ≡ 0 (mod 3) --> k ≡ 1 (mod 3)
2 + 2k ≡ 0 (mod 5) --> 1 + k ≡ 0 (mod 5) --> k ≡ 4 (mod 5)


The first two equations imply k ≡ 1 (mod 6), so k can be 1, 7, 13, 19, 25, 31, ... The first value of k that is 4 modulo 5 is 19, so the smallest term in the sequence that satisfies is 17 + 7(19), or 150.