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 $A + B?$

$7$

$8$

$11$

$14$

$15$

Answer: C

Concepts:place value divisibility modular arithmetic

Difficulty rating: 1390

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. $7A + 6B \equiv 0 \pmod{36}.$ Reducing once more modulo $6$ gives $A\equiv0\pmod6,$ so the only possible nonzero digit is $A=6.$ Then $42+6B$ is divisible by $36$ only for the digit $B=5.$ Indeed $\$655.56 = 36 \times \$18.21,$ so $A+B=11.$

Thus, the correct answer is C.

2025 AMC 12B Problem 6

Answer: C

Difficulty rating: 1390

Solution:

Problem 6 in Other Years