Question

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 $A \neq 0.$ " After a pause, Max says, "That was a good price." What is $A + B?$

$7$

$8$

$11$

$14$

$15$

Solution:

The total in cents is $10000A + 1110B + A = 10001A + 1110B,$ which must be a multiple of $36.$ Since $10001 \equiv 29$ and $1110 \equiv 30 \pmod{36},$ the condition is $29A + 30B \equiv 0,$ i.e. $7A + 6B \equiv 0 \pmod{36}.$ The only digit solution with $A \neq 0$ is $A = 6, B = 5$ ( $7 \cdot 6 + 6 \cdot 5 = 72$ ), giving $\$655.56 = 36 \times \$18.21.$ So $A + B = 11.$

Thus, the correct answer is C.

2025 AMC 12B Problem 6

Solution:

Problem 6 in Other Years