2017 AMC 12A Problem 18

Below is the professionally curated solution for Problem 18 of the 2017 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.

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Concepts:digits modular arithmetic

Difficulty rating: 1990

18.

Let $S(n)$ equal the sum of the digits of positive integer $n.$ For example, $S(1507)=13.$ For a particular positive integer $n,$ $S(n)=1274.$ Which of the following could be the value of $S(n+1)?$

$1$

$3$

$12$

$1239$

$1265$

Solution:

Adding $1$ to $n$ increases the digit sum by $1,$ except that each trailing $9$ turns into a $0,$ losing $9.$ If $n$ ends in exactly $k$ nines, then $S(n+1)=S(n)+1-9k=1275-9k.$

So the possible values are $1275,1266,1257,\ldots$ Among the choices, only $1239=1275-9\cdot4$ fits (for example, $n$ ending in four $9$ s preceded by enough $1$ s).

Thus, the correct answer is D.

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