Question

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

Answer 1

Any positive odd integer is of the form 2q + 1, where q is a whole number. 

Therefore, (2q + 1)² = 4q² + 4q + 1 = 4q (q + 1) + 1, ...... (1) 

q (q + 1) is either 0 or even. So, it is 2m, where m is a whole number. 

Therefore, (2q + 1)2 = 4.2 m + 1 = 8 m + 1. [From (1)]