Modular square root ($x^2\equiv a(\mathrm{mod}m)$)

A modular square root of a residue $a$ modulo $m$ is a residue $x$ with $x^2\equiv a(\mathrm{mod}m)$. Such a root need not exist or be unique. Modulo seven, $2$ has the two roots $3,4$, while $3$ has none.

For an odd prime $p$, each nonzero quadratic residue has exactly two roots, $x$ and $-x$, because $X^2-a$ has at most two roots in a field. A composite modulus can have more: $1$ has four roots modulo eight. For coprime factors of $m$, the Chinese remainder theorem combines local roots into all roots modulo $m$.