Square Numbers with 3s
Look at the following square numbers:
1, 4, 9, 16, 25, 36, ......
These square numbers give either a remainder of 0 or 1 when divided by 3. In this same fashion, is it possible for a square number to have a remainder of 2 when divided by 3?
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1 solution
Let x^2 be a square number. x either gives a remainder of 0, 1, or 2 when divided by 3. Consider the following cases:
Case 1: x = 0 (mod 3)
Then x^2 will still give 0^2 (mod 3) = 0 (mod 3).
Case 2: x = 1 (mod 3)
Then x^2 will give 1^2 (mod 3) = 1 (mod 3).
Case 3: x = 2 (mod 3)
Then x^2 will give 2^2 (mod3) = 4 (mod 3) = 1 (mod 3).
In all of the cases, x^2 will either be 0 (mod 3) or 1(mod 3). Hence it is impossible to get a result of 2 (mod 3) for x^2, so the answer is No.