Number of 1's and the number of 9's
The digit root of a number is obtained by repeatedly calculating the digit sum, until a single-digit number is reached. For example, the above illustrates that the digit root of 957107 is 2.
If we list out all the digit roots of the integers from 1 to (inclusive), what is the (non-negative) difference between the number of 1's and the number of 9's?
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As you might know, the sum of a numbers digits is used as a shortcut to see if a number is divisible by 9 or not.It can also be used to find the remainder when the original number is divided by 9.If the sum is 17, then the remainder is 8 because 17 mod 9=8 mod 9.Note that we could have also find the sum of the digits of 17 and repeated this until we get a number less than 10.This is exactly the same as what a digit root is, the sum of the digits of a number n and then the sum of the sum of the digits of n and so on.Thus, the digit root is just the same as the remainder! Since the numbers with remainder of 1 are 1, 10, 19... 1000, by counting, we can see that there are 112 numbers with a remainder of 1 while there are only 111 numbers with a remainder of 0 , this is that same as the digit root being 9, so the difference is 1.