2007 AMC 10A Problem 25

Problem 25 of 25HarderAlgebra Number Theory

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?

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Solution

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.

More practice

Concepts: digits · modular arithmetic · bounding to limit cases

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.