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smc

number theory senior

Problem

Let denote the sum of the digits of the positive integer . For example, and . For how many two-digit values of is ?
(A)
(B)
(C)
(D)
(E)
Solution
Let . Since , we have . Thus if , then or . The 3 values of for which are 12, 21, and 30, and the 7 values of for which are 39, 48, 57, 66, 75, 84, and 93. There are = values in all.
Final answer
E