Question

In the decimal system of numeration the number of $6$ - digits number in which the sum of the digits is divisible by $5$ is :

• A. $180000$
• B. $540000$
• C. $5\times {10}^5$
• D. None

Answer 1

The correct option is A $180000$

Since we have to make a 6 digit number, the first digit of the 6 digit number can be any digit except 0.

So, no. of choices for first digit$=9$

Digits at second,third,fourth and fifth position can be any digit w/o any constraints

So, no. of choices for second,third,fourth and fifth positions$=10each$

Now after placing first 5 digits and adding those digits, we will get the sum of first 5 digits.

Now if sum of those five digits has 1 in unit place then we can put 4 or 9 as the sixth digit.

Similarly, if sum has 2 in units place then we can use 3 or 8 as the sixth digit.

This implies, for every set of 5 digits we will have 2 options for the sixth digit.

So, total number of 6 digit numbers in which sum of digits is divisible by 5$=9\times 10\times 10\times 10\times 10\times 2$

$\Rightarrow 18\times {10}^4$

Hence, answer is option $A$