This was my reasoning for S2 without knowing the general rule about the unit digits you wrote about, I found the right answer but please tell me if the reasoning is flawed:
if (N(^2)+1) / 5 is an odd integer, then the nominator ( N^2 + 1 ) must be an odd multiple of 5 ( 15, 25, 35 ...). We know that N is an integer, so that odd muliple of 5 -1 must be an even perfect square square such as: N^2 = (odd multiple of 5) - 1= perfect even perfect square. From that : N^2 = 65 - 1 = 64 , N=8, plug it
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if (N(^2)+1) / 5 is an odd integer, then the nominator ( N^2 + 1 ) must be an odd multiple of 5 ( 15, 25, 35 ...). We know that N is an integer, so that odd muliple of 5 -1 must be an even perfect square square such as: N^2 = (odd multiple of 5) - 1= perfect even perfect square. From that : N^2 = 65 - 1 = 64 , N=8, plug it
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