Find the value of 2 / 3 in (mod 4)

We are looking for an integer $x$ such that when $3x$ is divided by 4, the remainder is 2.

$$3x \equiv 2 \pmod{4}$$

We need to find an integer $y$ such that $3y \equiv 1 \pmod{4}$.
Let's test values for $y$:

• If $y=0$, $3 \times 0 = 0 \not\equiv 1 \pmod{4}$
• If $y=1$, $3 \times 1 = 3 \not\equiv 1 \pmod{4}$
• If $y=2$, $3 \times 2 = 6 \equiv 2 \pmod{4}$
• If $y=3$, $3 \times 3 = 9 \equiv 1 \pmod{4}$ So, the multiplicative inverse of $3 \pmod{4}$ is 3.

Multiply both sides of $3x \equiv 2 \pmod{4}$ by 3:

$$3 \times (3x) \equiv 3 \times 2 \pmod{4}$$ $$9x \equiv 6 \pmod{4}$$

Since $9 \equiv 1 \pmod{4}$ and $6 \equiv 2 \pmod{4}$, substitute these values into the congruence:

$$1x \equiv 2 \pmod{4}$$ $$x \equiv 2 \pmod{4}$$

The value of $2 \div 3$ in $(\text{mod } 4)$ is $\boxed{\text{2}}$.