Show that square of an odd positive integer can be of the form 6q+1 or 6q+3 for some integer \'q\'

Show that square of an odd positive integer can be of the form 6q+1 or

1.	Show that square of an odd positive integer can be of the form 6q+1 or 6q+3 for some integer \'q\'
Answer» We know that any positive integer can be of the form 6m, 6m + 1, 6m + 2, 6m + 3, 6m + 4 or 6m + 5 for some integer m.So, an odd positive integer x is of the form 6m +1 or 6m + 3.CASE I When x = 6m + 1: In this case,x2 = (6m + 1)2 = 36m2 + 12m + 1 = 6 (6m2 + 2m) + 1 = 6q + 1, where q = 6m2 + mCASE II When x = 6m + 3: In this case,x2 = (6m + 3)2 = 36m2 + 36m + 9 = (36m2 + 36m + 6) + 3= 6 (6m2 + 6m + 1) + 3 = 6q + 3, where q = 6m2 + 6m + 3.