2007 AMC 10A Problem 25

Question

For each positive integer

n,

let

S(n)

denote the sum of the digits of

n.

For how many values of

n

is

n + S(n) + S(S(n)) = 2007?

1

2

3

4

5

For

n \le 2007,

S(n) \le 28

and

S(S(n)) \le 10,

so

n \ge 1969

n,

S(n),

and

S(S(n))

are all congruent modulo

9,

and

2007

is a multiple of

9

If

n \le 2007,

then

S(n) \le 28

and

S(S(n)) \le 10,

so

n \ge 2007 - 28 - 10 = 1969.

Since

n,

S(n),

and

S(S(n))

all leave the same remainder modulo

9

and

2007

is a multiple of

9,

each must be a multiple of

3.

Checking the multiples of

3

between

1969

and

2007,

the condition holds for

1977,

1980,

1983,

and

2001.

So there are

4

values of

n.

Thus, the correct answer is D.

Problem 25 in other years