Question

Number of $4$ digit positive integers, if the product of their digits is divisible by $3$, is

• A. $2700$
• B. $6628$
• C. $7704$
• D. $5464$

Answer 1

The correct option is C $7704$

We need to find $4$ digit numbers whose digits product is divisible by $3.$

At least one of the digit is divisible by $3.$

$\Rightarrow$ Required $=$ Total $-$ No digit is divisible by $3.$

$\Rightarrow$ Total number of $4$ digit numbers $=9\times 10\times 10\times 10$

$=9000.$

No digit is divisible by $3$ $\to 0,3,6,9$ must not appear.

$\therefore$ Numbers should be formed using only $1,2,4,5,7,8=6$ numbers.

$\Rightarrow$ No. of ways $=6\times 6\times 6\times 6=6^4=1296.$

$\Rightarrow$ Required $=9000-1296$

$=7704.$

Hence, the answer is $7704.$