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2025 AMC 10B Problem 8

Problem 8 of 25EasierNumber TheoryArithmetic

Emmy says to Max, “I ordered 3636 math club sweatshirts today.” Max asks, “How much did each shirt cost?” Emmy responds, “I’ll give you a hint. The total cost was $A‾ B‾ B‾.B‾ A‾,\$\underline{A}\,\underline{B}\,\underline{B}.\underline{B}\,\underline{A}, where AA and BB are digits and A≠0.A \ne 0.” After a pause, Max says, “That was a good price.” What is A+B?A + B?

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Solution

In cents the total is 10000A+1000B+100B10000A + 1000B + 100B +10B+A+ 10B + A =10001A+1110B.= 10001A + 1110B. Split evenly among 3636 shirts, so it’s divisible by 36.36. Now 10001≡2910001 \equiv 29 and 1110≡30(mod36),1110 \equiv 30 \pmod{36}, so we need 29A+30B≡0,29A + 30B \equiv 0, which reduces to 7A+6B≡0(mod36).7A + 6B \equiv 0 \pmod{36}. The only digit solution with A≠0A \ne 0 is A=6,B=5,A = 6, B = 5, since 7⋅6+6⋅5=72.7 \cdot 6 + 6 \cdot 5 = 72. That’s $655.56,\$655.56, or $18.21\$18.21 a shirt, so A+B=11.A + B = 11. Therefore, the answer is C.
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Tagged: divisibility · modular arithmetic · place value

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