2017 AMC 12A Problem 18

Attempt Problem 18 of the 2017 AMC 12A below, then check your answer against the professionally curated solution 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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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

n

ends in exactly

k

nines, adding

1

drops the digit sum by

9k

and adds

1

S(n+1)=1275-9k;

check which choice has this form

Adding

1

n

increases the digit sum by

1,

except that each trailing

9

turns into a

0,

losing

9.

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.

Problem 18 in other years