If $a _{1}, a _{2}, a _{3} \ldots$ and $b _{1}, b _{2}, b _{3} \ldots$ are $A.P.$ and $a_{1}=2, a_{10}=3, a_{1} b_{1}=1=a_{10} b_{10}$ then $a_{4} b_{4}$ is equal to

The number of terms of the $A.P. 3,7,11,15...$ to be taken so that the sum is $406$ is

The number of terms in the series $101 + 99 + 97 + ..... + 47$ is

If sum of $n$ terms of an $A.P.$ is $3{n^2} + 5n$ and ${T_m} = 164$ then $m = $

The sum of all the elements in the set $\{\mathrm{n} \in\{1,2, \ldots \ldots ., 100\} \mid$ $H.C.F.$ of $n$ and $2040$ is $1\,\}$ is equal to $.....$

If $1,\,\,{\log _9}({3^{1 - x}} + 2),\,\,{\log _3}({4.3^x} - 1)$ are in $A.P.$ then $x$ equals