The smallest $4-$digit number exactly divisible by $12,15,20$ and $35$ is .............

The correct option is C $1,260$
The smallest 4-digit $=1000$
Now L.C.M of $12,15,20and35$
$\underset{–}{2|12,15,20,35}$
$\underset{–}{2|6,15,20,35}$
$\underset{–}{3|15,5,35}$
$\underset{–}{5|1,5,5,35}$
$\underset{–}{7|1,1,1,1}$
1,1,1,1
$L.C.M.=2\times 2\times 3\times 5\times 7=420$
We divided $1000$ by $420$
$420\left)1000\right(2$
$\underset{–}{-840}$
$160-Remainder$
$\therefore$ The least $4-$digit number that is exactly divisible by $12,15,20$ and $35$
$=1000+(420-160)$
$=1000+260=1,260$