MathDB

Let

P_1,P_2,\ldots ,P_n

, be infinite arithmetic progressions of positive integers, of differences

d_1,d_2,\ldots ,d_n

, respectively. Prove that if every positive integer appears in at least one of the

n

progressions then one of the differences

d_i

divides the least common multiple of the remaining

n-1

differences.
Note:

P_i=\left \{ a_i,a_i+d_i,a_i+2d_i,a_i+3d_i,a_i+4d_i,\cdots \right \}

with

a_i

and

d_i

positive integers.

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