Square Modulo 4

Theorem

Let $x \in \Z$ be an integer.

Then $x$ is:

even if and only if $x^2 \equiv 0 \pmod 4$

odd if and only if $x^2 \equiv 1 \pmod 4$

Proof

\(\ds x\)

\(=\)

\(\ds 2 k\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x^2\)

\(=\)

\(\ds \paren {2 k}^2\)

\(\ds \)

\(=\)

\(\ds 4 k^2\)

\(\ds \)

\(\equiv\)

\(\ds 0\)

\(\ds \pmod 4\)

$\Box$

\(\ds x\)

\(=\)

\(\ds 2 k + 1\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x^2\)

\(=\)

\(\ds \paren {2 k + 1}^2\)

\(\ds \)

\(=\)

\(\ds 4 \paren {k^2 + k} + 1\)

\(\ds \)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod 4\)

$\blacksquare$

Also see

• Square Modulo 3
• Square Modulo 8

Sources

• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.1$ The Division Algorithm