Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.
I may begin saying that 2,3,4 belong to the 3 progressions indivudually.
edit: my working for above statement
let a1,a2,a3 be the 3 progressions now if a) all 3 belong to only one of them, then 1980 obviously exists in it b) if only two of 2,3,4 belong to one of them then also it becomes a series with common difference given so which means 1980 belongs to one of them
so without loss of generality, i can safely assume 2,3,4 to belong to each one of them indivudually i.e 2 to a1, 3 to a2 and 4 to a3. now how to proceed?
well i got this far due to a clue that was associated with the question.