The base-nine representation of the number N is 27,006,000,052_9. What is the remainder when N is divided by 5?

2021 AMC 12A Fall Problem 10

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10.

The base-nine representation of the number

N

27{,}006{,}000{,}052_9.

What is the remainder when

N

is divided by

5?

0

1

2

3

4

9 \equiv -1 \pmod 5,

9^k \equiv (-1)^k \pmod 5

Take the alternating sum of the base-nine digits (from the right)

Since

9 \equiv -1 \pmod 5,

each power

9^k \equiv (-1)^k,

N

is congruent to the alternating sum of its base-nine digits.

The nonzero digits, with their positions from the right, are

2

(position

0

5

(position

1

6

(position

6

7

(position

9

), and

2

(position

10

). The alternating sum is

2 - 5 + 6 - 7 + 2 = -2

\equiv 3 \pmod 5.

Thus, the correct answer is D.