Any odd positive integer is of the form 4q + 1 or 4q + 3 for some integer q. When n = 4q + 1, n²=(4q+1)² =16q² +8q+1 = 8q(2q+1)+1 =8m+1 where m = q(2q+1)
now n² is of the form 8m+1
WHEN n = 8m+3 n²=(4q+3)² = 16q²+9+ 24q = 16q²+24q+8+1 =8(2q²+3q+1)+1 =8m+1 where m = (2q²+3q+1)
n ² is of the form 8m+1 HENCE n² is of the form 8m +1 if n is an odd positive integer
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Verified answer
Any odd positive integer is of the form 4q + 1 or 4q + 3 for some integer q. When n = 4q + 1,
n²=(4q+1)²
=16q² +8q+1
= 8q(2q+1)+1
=8m+1 where m = q(2q+1)
now n² is of the form 8m+1
WHEN n = 8m+3
n²=(4q+3)²
= 16q²+9+ 24q
= 16q²+24q+8+1
=8(2q²+3q+1)+1
=8m+1 where m = (2q²+3q+1)
n ² is of the form 8m+1
HENCE
n² is of the form 8m +1 if n is an odd positive integer