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Suppose $a$ and $b$ are positive integers, neither of which is a multiple of 3 . Find the least possible remainder when $a^{2}+b^{2}$ is divided by $3 .$
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Since $a$ is not a multiple of 3 , then $a$ must be either 1 or 2 modulo 3 . Likewise, $b$ must be either 1 or 2 modulo 3 . Thus, we have four cases to consider:
1. If $a$ is 1 modulo 3 and $b$ is 1 modulo 3 , then $a^{2}$ is 1 modulo 3 and $b^{2}$ is 1 modulo 3 , so $a^{2}+b^{2}$ is 2 modulo 3 .
2. If $a$ is 1 modulo 3 and $b$ is 2 modulo 3 , then $a^{2}$ is 1 modulo 3 and $b^{2}$ is 1 modulo 3 , so $a^{2}+b^{2}$ is 2 modulo 3 .
3. If $a$ is 2 modulo 3 and $b$ is 1 modulo 3 , then $a^{2}$ is 1 modulo 3 and $b^{2}$ is 1 modulo 3 , so $a^{2}+b^{2}$ is 2 modulo 3 .
4. If $a$ is 2 modulo 3 and $b$ is 2 modulo 3 , then $a^{2}$ is 1 modulo 3 and $b^{2}$ is 1 modulo 3 , so $a^{2}+b^{2}$ is 2 modulo 3 .
Therefore, the least possible remainder is 2 . Final Answer: The final answer is 2 .
by Diamond (66,975 points)

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