Question

Show that every even positive integer is of the form 5m+1 5m+3 for some integer.

Answer 1

Let x be any positive integerThen x = 5q or x = 5q+1 or x = 5q+4 for integer x.If x = 5q, x2 = (5q)2 = 25q2 = 5(5q2) = 5n (where n = 5q2 )If x = 5q+1, x2 = (5q+1)2 = 25q2+10q+1 = 5(5q2+2q)+1 = 5n+1 (where n = 5q2+2q )If x = 5q+4, x2 = (5q+4)2 = 25q2+40q+16 = 5(5q2 + 8q + 3)+ 1 = 5n+1 (where n = 5q2+8q+3 )∴in each of three cases x2 is either of the form 5q or 5q+1 or 5q+4 and for integer q.