Question

The number of six-digit numbers which have sum of their digits as an odd integer, is

• A. $45000$
• B. $450000$
• C. $97000$
• D. $970000$

Answer 1

The correct option is B $450000$

The first six digit no.$=100000$
The last six digit no.$=999999$
No. of six digit no.$=999999-100000+1$
$=900000$

In every two consecutive numbers, one of them will have sum of their digits as even integer.

For example, $100000$ have sum of their digits equal to $1$

Then $100001$ will have sum of their digits equal to $2$ and so on...

So, every alternate number have sum of their digits as even number and similarly the other alternate pair of numbers have their sum of digits as odd integer.

Thus, the number of six-digit numbers which have sum of their digits as odd integer and the number of six-digit numbers which have sum of their digits as even integer are equal.

And we know that, sum of digits of the number is either odd or even.

Thus, the number of six-digit number having sum of their digits as odd integer is half of the total number we have
$\therefore$ Out of $900000$, half of them will have sum as odd.
Therefore $450000$ six digit no. will have sum of their digits as an odd integer.