Find the smallest number which, when increased by one is exactly divisible by $12,18,24,32$ and $40.$

Step: Find the smallest number

$L.C.M.$ of given numbers

$\therefore L.C.M.=2\times 2\times 2\times 2\times 2\times 3\times 3\times 5=1440$
$\therefore$ The smallest number $=1440-1$
$=1439$
Hence, the smallest number is $1439$.