The number of terms common to the two A.P.'s 3,7,11,,407 and 2,9,16,,709 is ________.

Question

The number of terms common to the two A.P.'s $3,7,11,\ldots,407$ and $2,9,16,\ldots,709$ is \_\_\_\_\_\_\_\_.

Answer: 14

Accepted Answer

Common terms form an A.P. with first term $23$ (first common term) and common difference $\mathrm{lcm}(4,7)=28$.

$$23+(n-1)28 \le 407 \implies n-1\le\frac{384}{28}=13.7 \implies n\le14.7 \implies n=14.$$

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