One way to look at this is to look at the mod 4 remainders of the cubes of 4 consecutive odd numbers.
1³ mod 4 = 1
3³ mod 4 = 3
5³ mod 4 = (5 mod 4)³ = 1³ = 1
7³ mod 4 = (7 mod 4)³ mod 4 = 3³ mod 4 = 3
The only possible remainders when the cube of an odd number is divided by 4 are 1 or 3.
