Prove that if $3|a^2+b^2\Rightarrow 3|a ,\, 3|b$

Question

How do i prove that if $$3|a^2+b^2\Rightarrow 3|a ,\, 3|b \,?$$ The book (Problem-Solving Strategies by Arthur Engels, page 43, E9) does not prove this theorem and leaves it as an excercise. I'm relatively new to this kind of problems and I just can't get any ideas. If anyone could prove that or, even better, just give me some hints it would be great. Thanks for your time

Hint: the number $a$ can be written as $a=3k+m$, where $k$ is an integer and $m$ is $0,1,$ or $2$. If you square $a$, we can further write $a^2=3k_2+m_2$ for new integers. If you know $m$, what is $m_2$? — Josh B., Mar 25 '21 at 19:46

score 1 · Accepted Answer · answered Mar 25 '21 at 19:46

1

Assume $a = 3A + r_1$ and $b = 3B + r_2$ where $r_1$ and $r_2$ are remainders. Then $$a^2+b^2 = 3C + r_1^2 + r_2^2$$ Therefore, if $3\mid a^2+b^2$, then $3 \mid r_1^2+r_2^2$. Since $r_1,r_2 \in \{0,1,2\}$, it is easy to verify that $r_1 = r_2 = 0$.

answered Mar 25 '21 at 19:46

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Please strive not to add more dupe answers to dupes of FAQs. – Bill Dubuque Mar 25 '21 at 19:47

Prove that if $3|a^2+b^2\Rightarrow 3|a ,\, 3|b$

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