2025 AMC 10B Problem 8

Problem 8 of 25EasierNumber Theory

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?

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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

A = 6, B = 5,

since

7 \cdot 6 + 6 \cdot 5 = 72.

That’s

\$655.56,

\$18.21

a shirt, so

A + B = 11.

Therefore, the answer is C.