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Consider that $$[p_n]$$ and $$[qn]$$ are two arithmetic progressions with 50 elements each. If $$[p_n]$$ has the elements $$a_1 = 4, a_2 = 6$$ and so on, and $$[qn]$$ has $$b_1 = 2, b_2 = 5$$ and so on, how many common elements do both $$[p_n]$$ and $$[qn]$$ have ?
The common difference of $$[p_n]$$ is 2, while that of $$[qn]$$ is 3.
The last term for $$[p_n]$$ will be 4 + 49$$\times\ $$2, which is 102
Similarly, the last term for $$[qn]$$ will be 2 + 49$$\times\ $$3, which is 149
Now, the first common term for these two series is 8. Further, the common difference for the series of common terms shall be the LCM (2,3), which is 6.
Hence, 8 + 6K < 102
Or, K < 15.666....
This means K can take the maximum value of 15 and minimum of 0. Hence, 16 terms are common to the two series.
