# Progressions| Existence of 1980 in the given progression

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.

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I see what you mean, but can you include a reasoning for this claim. In general, it is a nice idea to show your work so that others can see where you are stuck. (Also, please avoid posting questions in the imperative mode.) –  Srivatsan Sep 3 '11 at 14:08