2007 AMC 12A Problem 22

Problem 22 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

For

n\le 2007,

S(n)\le S(1999)=28,

and then

S(S(n))\le S(28)=10.

So any solution has

n\ge 2007-28-10=1969.

Also

n,

S(n),

and

S(S(n))

are congruent modulo

9,

and

2007

is a multiple of

9,

so all three must be multiples of

3.

Checking the multiples of

3

between

1969

and

2007

(many are eliminated because

n+S(n)

already exceeds

2007

) leaves

1977,1980,1983,

and

2001.

That is

4

values. 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.