🔥 New! 2025 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 8 Easy

Emmy says to Max, “I ordered 36 math club sweatshirts today." Max asks, “How much did each shirt cost?" Emmy responds, “I'll give you a hint. The total cost was \$ \underline A \underline B \underline B. \underline B \underline A, where A and B are digits and A\ne 0." After a pause, Max says, “That was a good price." What is A+B?

  • A.

    7

  • B.

    8

  • C.

    11

  • D.

    14

  • E.

    15

Answer:C
Solution

Given 36 \mid (ABBBA)_{10}, we need 9 \mid 3B + 2A and 4 \mid (BA)_{10}

From 3 \mid 3B + 2A, we get 3 \mid A. Since (BA)_{10} is even, A is even, so 2 \mid A. Thus, A = 6.

From 9 \mid 3B + 2A = 3B + 12, we get 3 \mid B + 4, so B \in \{2, 5, 8\}.

From 4 \mid 10B + A = 10B + 6, we get 2 \mid 5B + 3, which means B is odd.

Therefore B = 5.