2017 AMC 12A Problem 18

Problem 18 of 25IntermediateNumber Theory

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)?

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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.

More practice

Concepts: digits · modular arithmetic

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