Question

How many four digit number are divisible by 5, 12 and 18

• A. 48
• B. 49
• C. 47
• D. 50

Answer 1

The correct option is D 50

$\Rightarrow$ $5=1\times 5=5$

$\Rightarrow$ $12=2\times 2\times 3=2^2\times 3$

$\Rightarrow$ $18=2\times 3\times 3=2\times 3^2$

$\Rightarrow$ L.C.M of 5, 12 and 18 = $2^2\times 3^2\times 5=180$

$\Rightarrow$ So our four digit number must be divisible by $180.$

$\Rightarrow$ Smallest four digit number = 1000

$\Rightarrow$ So, $\frac{1000}{180}=5.55\approx 6$

$\Rightarrow$ Smallest four digit number that is divisible by 180 = $180\times 6=1080$

$\Rightarrow$ We know highest four digit number = $9999$

$\Rightarrow$ So, $\frac{9999}{180}=55.55\approx 55$

$\therefore$ Highest four digit number that is divisible by 180 = $180\times 55=9900$

$\Rightarrow$ So, $1080,1260,....9900$ is an AP.

$\Rightarrow$ Here, $a_1=1080,d=180,a_n=9900$

$\therefore$ $a_n=a_1+(n-1)d$

$\therefore$ $9900=1080+(n-1)180$

$\therefore$ $9900=1080+180n-180$

$\therefore$ $9900=900+180n$

$\therefore$ $180n=9000$

$\therefore$ $n=50$.

$\therefore$ There are $50$ four digit number which are divisible by 5, 12 and 18.