If $3|(a^2 + b^2)$, show that $3|a$ and $3|b$.

Question

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If $3|(a^2 + b^2)$, show that $3|a$ and $3|b$.

Answer 1

The key point is that if $3$ does not divide $x$, then $x^2$ leaves remainder $1$, because $x$ is of the form $3k\pm1$. Therefore:

If $3$ divides $a$ but not $b$ (or vice-versa), then $a^2+b^2$ leaves remainder $1$.

If $3$ divides neither $a$ nor $b$ then $a^2+b^2$ leaves remainder $2$.

So, the only way for $3$ to divide $a^2+b^2$ is for $3$ to divide both numbers.

Answer 2

From $3\mid a^2+b^2$ we get $a^2\equiv -b^2\pmod 3$. Now assume that $3\nmid a,b$, then because $3$ is prime we get $\gcd(3,a)=\gcd(3,b)=1$. Hence we may divide the first congruence by either $a$ or $b$ (let's do $b$) to get $(a/b)^2\equiv -1\pmod 3$. Hence $-1$ is a quadratic residue mod $3$, contradiction. Thus $3\mid a,b$.

Answer 3

Say $a = 3k_1 + i_1$ and $b = 3k_2 + i_2$, where $0\leq i_1, i_2< 3$. Then we have $$ a^2 = (3k_1^2 + i_1)^2 = 9k_1 + 6i_1k_1 + i_1^2 = 3(3k_1^2 + 2 k_1i_1) + i_1^2 $$ and similarily, we have $$ b^2 = 3(3k_2^2 + 2k_2i_2) + i_2^2 $$ Now, each of $i_1$ and $i_2$ are either $0, 1$ or $4$, and you can verify by checking each case that if one or both of them are not $0$, then $i_1^2 + i_2^2$ is not divisible by $3$.

Answer 4

The quadratic residues mod $3$ are $0$ and $1$. The only sum of two of these that is $0$ mod $3$ is $0+0$. Therefore, $a^2\equiv0\pmod3\implies a\equiv0\pmod3$ and $b^2\equiv0\pmod3\implies b\equiv0\pmod3$.