Average digit sum – 5

Number Theory Level 5

Let $s(n)$ be the digit sum of $n$ .

$d_a(n)$ is defined to be the average digit sum of all non-negative integers less than or equal to $n$ that are divisible by $a$ .

$d_a(n)= \displaystyle \frac 1{\left\lfloor \frac na \right\rfloor +1} \sum_{k=0}^{\left\lfloor \frac na \right\rfloor} s(ak)$

Find $d_{9}\left(10^8\right) - d_{11}\left(10^8\right)$ .

Other parts

Average digit sum – 6
Average digit sum – 4

The answer is 0.

1 solution

Chris Lewis
Mar 4, 2019

We can use the same approach as for the previous problem.

If $x$ is a multiple of $9$ , so is $99999999-x$ . If both members of this pair are positive, then the sum of their digit sums will always be $72$ . It follows that $d_{9} \left(10^8 \right)$ is half of this, or $36$ .

In exactly the same way, we find that $d_{11} \left(10^8 \right)$ is also $36$ . Hence the required answer is $\boxed0$ .

This trick will work for any even power of $10$ .

Average digit sum – 5

The answer is 0.

1 solution

0 pending reports