Question

show that the cube of any positive integer is either of the form 2M OR 2M+1 FOR SOME INTEGER.

Answer 1

Hi!

Here is the answer to your question.

Let a be any positive integer. So, a is either an even positive integer or an odd positive integer.

∴ a = 2n or a = 2n + 1

a³ = (2n)³ = 8n³ = 2(4n³) = 2m, where m = 4n³

Or a³ = (2n + 1)³ = 8n³ + 12n² + 6n + 1 = 2(4n³ + 6n² + 3n) + 1 = 2m + 1, where m = 4n³ + 6n² + 3n

So, the cube of any positive integer is either of the form 2m or 2m+ 1, where m is integer.

Cheers!