Why is division by 2 possible modulo 5 but impossible modulo 4?

In modular arithmetic, division by a number $s$ modulo $N$ is defined as finding a value $X$ such that $s\times X\equiv 1\left(\mathrm{mod}N\right)$.

Division by 2 is possible modulo 5 because 2 and 5 are relatively prime—meaning their only common factor is 1. Specifically, $2\times 3=6$, and since $6\equiv 1\left(\mathrm{mod}5\right)$, the value 3 acts as the multiplicative inverse of 2, making $1÷2\equiv 3\left(\mathrm{mod}5\right)$.

Conversely, division by 2 is impossible modulo 4 because 2 and 4 are not relatively prime; they share a common factor of 2. For any integer $X$, the product $2\times X$ will always be an even number. In modulo 4, the possible values for $2\times X$ are $2\times 0\equiv 0$, $2\times 1\equiv 2$, $2\times 2\equiv 0$, and $2\times 3\equiv 2$. Because none of these products result in 1, there is no solution for $X$ in the equation $2\times X\equiv 1\left(\mathrm{mod}4\right)$.1