# Digit Extrema!

**Number Theory**Level 4

Let \(A_N\) denote the set of digit sums of \(N\)-digit numbers. Let \(B_N\) be the set of the digit sums of elements of \(A_N\). Let \(b_N\) be the largest element of \(B_N\). Let \(a_N\) be the largest element of \(A_N\) which gives \(b_N\) as a digit sum. Find \(a_{10}\)

For example, \(A_1={1, 2, 3, 4, 5,\ldots, 9}\), and \(B_1={1,2,3,4,5,\ldots,9}\). Thus, \(b_1=9\), an d \(a_1=9\)