Question

The number of terms common to the two arithmetic progressions
$3,7,11,\dots ,407$ and $2,9,16,\dots ,709$ is

Answer 1

Given A.P.s are
$\to 3,7,11,\dots ,407$ (common difference is $4$)
$\to 2,9,16,\dots ,709$ (common difference is $7$)
We can observe that the first common term is $23$.
( $6^{th}$ term in $1^{st}$ A.P. and $4^{th}$ term in $2^{nd}$ A.P. are same.)
Let the number of common terms be $n$.
They will be in A.P. with common difference $=28$ ( i.e., (LCM $(4,7)$ )
Now, the last common term can't be greater than $407$.
$\Rightarrow 23+(n-1)28\le 407$
$\Rightarrow n\le \frac{384}{28}+1$
$\Rightarrow n\le 14.7\Rightarrow n=14$