Algebra Solutions: Compute the possible residues of odd squares mod 8

Prove that the square of any odd integer always leaves a remainder of 1 when divided by 8.

We need to find the residues of square odd numbers mod 8. The odds mod 8 are precisely $\overline{1}$ , $\overline{3}$ , $\overline{5}$ , and $\overline{7}$ , and we have $\overline{1}^2 = \overline{1}$ , $\overline{3}^2 = \overline{9} = \overline{1}$ , $\overline{5}^2 = \overline{25} = \overline{1}$ , and $\overline{7}^2 = \overline{49} = \overline{1}$ .

Algebra Solutions

Compute the possible residues of odd squares mod 8

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