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Math Help - Prove 3 divides two integers

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    Prove 3 divides two integers

    If b and c are integers and 3 divides b^2 + c^2, prove that 3 divides both b and c.

    Since 3 divides b^2 + c^2, that means b^2 + c^2 ≡ 0 mod 3. Since b and c are squared (and 0^2 ≡ 0, 1^2 ≡ 2^2 ≡ 1 mod 3) the only way this can happen is if b ≡ c ≡ 0 mod 3. This means b and c are multiple of 3, so 3 divides both b and c. Is this correct reasoning?
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    Quote Originally Posted by IronicMan View Post
    If b and c are integers and 3 divides b^2 + c^2, prove that 3 divides both b and c.

    Since 3 divides b^2 + c^2, that means b^2 + c^2 ≡ 0 mod 3. Since b and c are squared (and 0^2 ≡ 0, 1^2 ≡ 2^2 ≡ 1 mod 3) the only way this can happen is if b ≡ c ≡ 0 mod 3. This means b and c are multiple of 3, so 3 divides both b and c. Is this correct reasoning?
    Yes; and if you wanted to be more explicit you could create a table (mod 3)

    0 + 0 = 0
    0 + 1 = 1
    (1 + 0 = 1)
    1 + 1 = 2
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