Square Modulo 3/Corollary 1

Corollary to Square Modulo 3

Let $x, y \in \Z$ be integers.

Then:

$3 \divides \paren {x^2 + y^2} \iff 3 \divides x \land 3 \divides y$

where $3 \divides x$ denotes that $3$ divides $x$.

Proof

Sufficient Condition

Let $3 \divides x \land 3 \divides y$.

Then by definition of divisibility:

$x \equiv 0 \pmod 3$

and

$y \equiv 0 \pmod 3$

From Square Modulo 3:

$x^2 \equiv 0 \pmod 3$

and

$y^2 \equiv 0 \pmod 3$

Then from Modulo Addition is Well-Defined:

$\paren {x^2 + y^2} \equiv 0 \pmod 3$

Thus:

$3 \divides \paren {x^2 + y^2}$

$\Box$

Necessary Condition

Now suppose $3 \divides \paren {x^2 + y^2}$.

Then by definition of divisibility:

$\paren {x^2 + y^2} \pmod 3$

From Square Modulo 3:

$x^2 \equiv 0 \pmod 3$ or $x^2 \equiv 1 \pmod 3$

and

$y^2 \equiv 0 \pmod 3$ or $y^2 \equiv 1 \pmod 3$

Thus the only way $\paren {x^2 + y^2} \equiv 0 \pmod 3$ is for:

$x^2 \equiv 0 \pmod 3$

and:

$y^2 \equiv 0 \pmod 3$

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Some Properties of $\Z$: Exercise $2.9$