Solution
\[
\mathrm{AP}_{1} \text { Set } \mathrm{A}=\{1,6,11,16 \ldots \ldots \ldots \ldots\}, \mathrm{n}(\mathrm{~A})=101
\]
\[
\begin{aligned}
& \mathrm{T}_{101}^{\mathrm{A}}=1+100 \times 5=501 \\
& \mathrm{AP}_{2} \text { Set } \mathrm{B}=\{9,16 \ldots \ldots \ldots\} \mathrm{N}(\mathrm{~B})=71 \\
& \mathrm{~T}_{71}^{\mathrm{B}}=9+70 \times 7=499
\end{aligned}
\]
Common A.P., common difference $\mathrm{D}=\mathrm{L} . \mathrm{C} . \mathrm{M}\left\{\mathrm{d}_{1}, \mathrm{~d}_{2}\right\}=35$;
$1^{\text {st }}$ Common term is 16
\[
\begin{aligned}
& 16+(n-1) 35 \leq 499 \Rightarrow n \leq 14.8 \Rightarrow n=14 \\
& T_{n}^{A \cap B}=16+(n-1) 35=35 n-19 \text { is divisible by } 3 \\
& =3(11 n-6)+2 n-1 \text { is divisible by } 3 \\
& 2 n-1 \text { must be divisible by } 3 \\
& n \in\{2,5,8,11,14\} \therefore 5 \text { such terms }
\end{aligned}
\]
[CHAPTER M08 PERMUTATION \& COMBINATION TOPIC T3 ]
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