We define the digit sum of a non-negative integer to be the sum of its digits. For example, the digit sum of 123 is $1 + 2 + 3 = 6$. (i) How many positive integers less than 100 have digit sum equal to 8 ? Let $n$ be a positive integer with $n < 10$. (ii) How many positive integers less than 100 have digit sum equal to $n$ ? (iii) How many positive integers less than 1000 have digit sum equal to $n$ ? (iv) How many positive integers between 500 and 999 have digit sum equal to 8 ? (v) How many positive integers less than 1000 have digit sum equal to 8 , and one digit at least 5 ? (vi) What is the total of the digit sums of the integers from 0 to 999 inclusive?
& 4 or 6
\section*{5. For ALL APPLICANTS.}
We define the digit sum of a non-negative integer to be the sum of its digits. For example, the digit sum of 123 is $1 + 2 + 3 = 6$.\\
(i) How many positive integers less than 100 have digit sum equal to 8 ?
Let $n$ be a positive integer with $n < 10$.\\
(ii) How many positive integers less than 100 have digit sum equal to $n$ ?\\
(iii) How many positive integers less than 1000 have digit sum equal to $n$ ?\\
(iv) How many positive integers between 500 and 999 have digit sum equal to 8 ?\\
(v) How many positive integers less than 1000 have digit sum equal to 8 , and one digit at least 5 ?\\
(vi) What is the total of the digit sums of the integers from 0 to 999 inclusive?