2021 Fall AMC 10A Problem 12

Problem 12 of 25IntermediateNumber Theory

The base-nine representation of the number

N

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

What is the remainder when

N

is divided by

5?

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Solution

Note that

9 \equiv -1 \pmod{5}.

Then if expand

N

using the definition of bases, we get

N = 2 \cdot 9^{10} + 7 \cdot 9^9 + 6 \cdot 9^6 +

5 \cdot 9 + 2

\equiv 2 (-1)^{10} + 7 (-1)^9 + 6 (-1)^6 +

5 (-1) + 2 \pmod{5}

\equiv 2 - 7 + 6 - 5 + 2 \pmod{5}

\equiv 3 \pmod{5}.

Thus, D is the correct answer.