Thread: 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?

Originally Posted by IronicMan

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