Find the smallest $4-$ digit number which is divisible by $18,24$ and $32$

Smallest number divisible by $18,24,32$

Factors of $18=2\times 3\times 3$

Factors of $24=2\times 3\times 2\times 2$

Factors of $32=2\times 2\times 2\times 2\times 2$

LCM$=2\times 2\times 2\times 2\times 2\times 3\times 3=288$

$\therefore 288$ is the smallest number divisible by $18,24,32$

Now, smallest $4$ digit number$=1000$

$1000=288\times 3+136$

Remainder$=136$

But we need to find a number where the remainder is $0$

Required number$=1000+(288-136)=1000+152=1152$

So,$1152$ is the smallest $4$ digit number.