Question

Show that cube of any positive integer is either of the form 4m, 4m + 1 or 4m + 3 for some integer m.

Answer 1

Let x is a positive integer.

Applying Euclid division lemma it is in the form of 4q or 4q + 1 or 4q + 1 or 4q + 2 or 4q + 3.

Now following cases are

where m = 16q³ + 36q² + 27q + 6

Hence cube of any positive integer is either  of the form 4m, 4m + 1, or 4m + 3 for some integer.