Question

The number of $5$-digit numbers that can be formed using the digits $0,2,3,6,8,9$ (without repetition) such that it is divisible by $4$ is

• A. $196$
• B. $144$
• C. $216$
• D. $204$

Answer 1

The correct option is D $204$

Divisibility by $4:$
The last two digits should be divisible by $4$.

So, the possible last two digits are,
$\left(08\right),\left(20\right),\left(28\right),\left(32\right),\left(36\right),\left(60\right),\left(68\right),\left(80\right),\left(92\right),\left(96\right)$

Case $1:$
When the last two digits are $\left(08\right),\left(20\right),\left(60\right),\left(80\right)$
Number of such $5$-digit numbers is
$4\times 3\times 2\times 4=96$

Case $2:$
When the last two digits are $\left(28\right),\left(32\right),\left(36\right),\left(68\right),\left(92\right),\left(96\right)$
Number of such $5$-digit numbers is
$3\times 3\times 2\times 6=108$

Hence, total numbers of $5$-digit numbers divisible by $4$ is $96+108=204$