Library / HMMT February Guts Round /1 of 28

HMMT February Guts Round, 2024

Number theory Difficulty 4.6 AIME Find the answer United States

Problem:
Let $a \star b = a b - 2$ . Compute the remainder when $(((579 \star 569) \star 559) \star \cdots \star 19) \star 9$ is divided by $100$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Note that
$(10 a + 9) \star (10 b + 9) = (100 a b + 90 a + 90 b + 81) - 2 \equiv 90(a + b) + 79 \pmod{100}$
so throughout our process all numbers will end in $9$ , so we will just track the tens digit. Then the "new operation" is
$a \dagger b \equiv -(a + b) + 7 \pmod{10}$
where $a$ and $b$ track the tens digits. Now
$(a \dagger b) \dagger c \equiv (-(a + b) + 7) \dagger c \equiv a + b - c \pmod{10}.$
Thus, our expression has tens digit congruent to
$-0 + 1 - 2 + 3 - \cdots - 54 + 55 - 56 - 57 + 7 \equiv -28 - 57 + 7 \equiv 2 \pmod{10}$
making the answer $29$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.