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

Question

Professor PC Thomas Classes · Answer

(14) : Given A.P.'s are $3,7,11, \ldots ., 407$ and $2,9,16, \ldots, 709$.\ Since, the given, A.P.'s have common difference 4 and 7 respectively. So, first common term to both A.P.'s $=23$\ Now, common difference of A.P.'s of common terms to given A.P.'s is L.C.M. $(4,7)=7 	imes 4$ $=28$\ Last term $\leq 407$\ $\Rightarrow \quad 23+(n-1) 	imes 28 \leq 407$\ $\Rightarrow \quad(n-1) 	imes 28 \leq 384$\ $\Rightarrow \quad n \leq 13.71+1 \Rightarrow n \leq 14.71$. So, $n=14$\