Suppose n is odd. find the cube of n and divide it by 4. what possible remainders could occur? check all the possible ones and none of the impossible ones.
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.
