2025 AMC 12B Problem 6

Problem 6 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 \neq 0.

” After a pause, Max says, “That was a good price.” What is

A + B?

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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,

A+B=11.

Thus, the correct answer is C.