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

2021 AMC 12A Fall Problem 10 — Solution

2021 AMC 12A Fall Problem 10

Below is the professionally curated solution for Problem 10 of the 2021 AMC 12A Fall, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Fall solutions, or check the answer key.

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Concepts:modular arithmetic number base

Difficulty rating: 1560

10.

The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\text{nine}}.$ What is the remainder when $N$ is divided by $5?$

$0$

$1$

$2$

$3$

$4$

Solution:

Since $9 \equiv -1 \pmod 5,$ each power $9^k \equiv (-1)^k,$ so $N$ 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.

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