Question

Number of 4 digit positive integer, if the product of their digit is divisible by 3, is
\[\begin{align}
  & A.2700 \\
 & B.6628 \\
 & C.7704 \\
 & D.5964 \\
\end{align}\]

Answer 1

Hint: In this question, we have to find number 4 digit positive integers in which product of their digit is divisible by 3. For this, we will first find a total 4 digit number. Since, we need the product to be divisible by 3, so there should be at least one of the four digits that is divisible by 3. For easily calculating, we will find numbers who do not have any digit divisible by 3 and subtract it from the total number of 4 digit numbers to get our required count.

Complete step by step answer:
Here, we need 4 digit numbers whose product is divisible by 3. So, we need to find numbers whose at least one digit is divisible by 3. But it is difficult to find such numbers, so we will find numbers whose none of the digits are divisible by 3. After that, we will subtract it from the total number of 4 digit numbers.
Now let us find the total number of 4 digit numbers. We need 4 places to fill and we have 10 digits. For one’s place, any of the 10 digits can take place, for hundreds places any of the 10 digits can take place. But for thousands of places, we cannot have digit 0 as it will change to a 3 digit number, so we have 9 digits that can take place.
Hence, total 4 digit numbers are $9\times 10\times 10\times 10=9000$.
Now, let us find a number who does not have any digit divisible by 3. As we know, numbers divisible by 3 are 0, 3, 6, 9. So, we cannot use these numbers to form our four digit number. Hence, we are left with 1, 2, 4, 5, 7 and 8 = 6 numbers.
So, all places (ones, tens, hundreds, thousands) can have any of the six digits. Therefore, such numbers are $6\times 6\times 6\times 6=1296$.
Hence, there are 1296 numbers where product of digits is not divisible by 3.
Therefore, numbers whose product of digit is divisible by 3 will be equal to total four digit numbers - number whose products of digits is not divisible by 3.
Hence, required numbers are: $\Rightarrow \text{9}000-\text{1296 }=\text{ 77}0\text{4}$.

So, the correct answer is “Option C”.

Note: Students should keep in mind that, we cannot take 0 in the thousands place while counting four digit numbers. Students can make mistakes of forgetting 0 as one of the numbers divisible by 3. Students should take care while subtracting large numbers.