🔥 New! 2025 AMC 10 B

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Problem 4 Easy

The value of the two-digit number \underline a \underline b in base seven equals the value of the two-digit number \underline b \underline a in base nine. What is a+b?

  • A.

    7

  • B.

    9

  • C.

    10

  • D.

    11

  • E.

    14

Answer:A
Solution

(ab)_7 = (ba)_9 implies 7a + b = 9b + a where 0 \leq a, b < 7

This gives 6a = 8b, then 3a = 4b, so a = 4k and b = 3k for k \in \mathbb{Z}.

Since a \in (0, 7), we have k = 1, thus a = 4 and b = 3.

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