To find the number of $10$ digit numbers, where the sum of digits is divisible by $10$.
The sum of digits is divisible by $10$ implies the sum has to be a multiple of $10$.
Since the number is $10$ digited, the first term has to take values greater than $0$.
Some of the numbers are all ones, $11\cdots11, 12\cdots10, 13\cdots100$.
I am finding it difficult to count the number of such numbers.
Best Answer
Since we have to choose a $10$ digit number the first position has $9$ choices, then from the second position to the last but one we have $10$ choices each.
Now we sum all these $9$ entries and choose the last digit such that the sum is divisible by $10$. We can always do it and the choice for the last position is unique.
Hence the total number of $10$ digit numbers, where the sum of digits is divisible by $10$ is $$9 \times 10^8. $$