Question

Show that the square of odd positive integers is of the form 8m+1 for some whole number m .

Answer 1

Any odd positive number can be denoted as 2n + 1 where n = 0, 1, 2, ... Hence, the square of any odd positive integer = (2n +1)^2 = 4n^2 + 4n + 1 = 4n(n + 1) + 1 Except for the case when n = 0, the above expression can be written as 4*2m + 1 (Since, n(n +1) = 2m for n, m being positive integers) or, 8m + 1. Thus, (2n + 1)^2 = 8m + 1 is shown.