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Philippine Mathematical Olympiad

Number theory Difficulty 4.9 AIME Find the answer Philippines

Problem:

Consider addition $\oplus$ and multiplication $\otimes$ modulo 7 of the numbers in $S=\{0,1,2,3,4,5,6\}$ . This means that
$\begin{aligned} & m \oplus n=\text{ remainder when } m+n \text{ is divided by } 7 \\ & m \otimes n=\text{ remainder when } m \times n \text{ is divided by } 7 \text{. } \end{aligned}$
Then 1 is the multiplicative identity and each element $a \in S$ has a multiplicative inverse $\frac{1}{a}$ . Find the value of $\frac{1}{4} \oplus\left(2 \otimes \frac{1}{3}\right)$ in this number system.

Your answer

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

Solution

Solution:

First, we need to find $\frac{1}{4}$ and $\frac{1}{3}$ in $S$ under multiplication modulo 7.

$\frac{1}{4}$ is the number $x$ such that $4 \otimes x \equiv 1 \pmod{7}$ .
Try $x = 2$ : $4 \times 2 = 8 \equiv 1 \pmod{7}$ , since $8 - 7 = 1$ .
So $\frac{1}{4} = 2$ .

$\frac{1}{3}$ is the number $y$ such that $3 \otimes y \equiv 1 \pmod{7}$ .
Try $y = 5$ : $3 \times 5 = 15 \equiv 1 \pmod{7}$ , since $15 - 2 \times 7 = 1$ .
So $\frac{1}{3} = 5$ .

Now compute $2 \otimes \frac{1}{3} = 2 \otimes 5 = 10 \equiv 3 \pmod{7}$ .

Now compute $\frac{1}{4} \oplus (2 \otimes \frac{1}{3}) = 2 \oplus 3 = 5$ (since $2 + 3 = 5 < 7$ ).

Therefore, the value is $\boxed{5}$ .

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