Question

The number of six-digit numbers (repetition is allowed) whose sum of the digits is divisible by $5$ is

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

Answer 1

Answer: (A)

$180000$
First place from left cannot be filled with $0$.
Next four places can be filled with any of the $10$ digits.

After filling the first five places, the last place can be filled in following ways.

CASE I : If the sum of first $5$ digits is of $5k$ type, then rightmost digit would be either $0\text{or}5$, so units place can be filled in $2$ ways

CASE II : If the sum of first $5$ digits is of $5k+1$ type, then rightmost digit would be either $4\text{or}9$, so units place can be filled in $2$ ways

CASE III : If the sum of first $5$ digits is of $5k+2$ type, then rightmost digit would be either $3\text{or}8$, so units place can be filled in $2$ ways

CASE IV : If the sum of first $5$ digits is of $5k+3$ type, then rightmost digit would be either $2\text{or}7$, so units place can be filled in $2$ ways

CASE : If the sum of first $5$ digits is of $5k+4$ type, then rightmost digit would be either $1\text{or}6$, so units place can be filled in $2$ ways


$\therefore$ The total such numbers possible
$=9\times {10}^4\times 2=180000$