Let a be any positive integer.
Then a = 8m + 1 where m is some integer
a = 8m + 1
⟹ a2 = (8m + 1)2
= 64m2 + 16m + 1
= 8q + 1
Where q = 8m2 + 2m
Hence square of an odd integer is of the form 8q + 1 where q is a positive integer.
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