# Difference between revisions of "2017 AMC 10A Problems/Problem 20"

## Problem

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 $S(n+1)$?

$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$

## Solution 1

Note that $n \equiv S(n) \pmod{9}$. This can be seen from the fact that $\sum_{k=0}^{n}10^{k}a_k \equiv \sum_{k=0}^{n}a_k \pmod{9}$. Thus, if $S(n) = 1274$, then $n \equiv 5 \pmod{9}$, and thus $n+1 \equiv S(n+1) \equiv 6 \pmod{9}$. The only answer choice that is $6 \pmod{9}$ is $\boxed{\textbf{(D)}\ 1239}$.

## Solution 2

We can find out that the least number of digits the number $N$ is $142$, with $141$ $9$'s and one $5$. By randomly mixing the digits up, we are likely to get: $9999$...$9995999$...$9999$. By adding $1$ to this number, we get: $9999$...$9996000$...$0000$. Knowing that this number is ONLY divisible by $9$ when $6$ is subtracted, we can subtract $6$ from every available choice, and see if the number is divisible by $9$ afterwards. After subtracting $6$ from every number, we can conclude that $1233$ (originally $1239$) is the only number divisible by $9$. So our answer is $\boxed{\textbf{(D)}\ 1239}$.

~ProGameXD