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

$7$

$8$

$11$

$14$

$15$

Solution:

In cents the total is $10000A + 1000B + 100B + 10B + A = 10001A + 1110B.$ Split evenly among $36$ shirts, so it's divisible by $36.$ Now $10001 \equiv 29$ and $1110 \equiv 30 \pmod{36},$ so we need $29A + 30B \equiv 0,$ which reduces to $7A + 6B \equiv 0 \pmod{36}.$ The only digit solution with $A \ne 0$ is $A = 6, B = 5,$ since $7 \cdot 6 + 6 \cdot 5 = 72.$ That's $\$655.56,$ or $\$18.21$ a shirt, so $A + B = 11.$ Therefore, the answer is C.

2025 AMC 10B Problem 8

Solution:

Problem 8 in Other Years