Let (x) denote the sum of the digits of the positive integer x. For example, (8) = 8 and (123) = 1 + 2 + 3 = 6. For how many two-digit values of x is ((x)) = 3?

2003 AMC 12B Problem 8

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Let

\clubsuit(x)

denote the sum of the digits of the positive integer

x.

For example,

\clubsuit(8) = 8

and

\clubsuit(123) = 1 + 2 + 3 = 6.

For how many two-digit values of

x

\clubsuit(\clubsuit(x)) = 3?

3

4

6

9

10

Let

y = \clubsuit(x).

For a two-digit

x,

y

is between

1

and

18

\clubsuit(y) = 3

forces

y = 3

y = 12;

count two-digit

x

with each digit sum

Let

y = \clubsuit(x).

Since

x \le 99,

we have

y \le 18.

Then

\clubsuit(y) = 3

requires

y = 3

y = 12.

The two-digit numbers with digit sum

3

are

12, 21, 30

(

3

values), and those with digit sum

12

are

39, 48, 57, 66, 75, 84, 93

(

7

values), for

10

in all.

Thus, the correct answer is E.