Show that any positive odd integer …

Let x be any positive integer
Then 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 )
<font face="Cambria Math, serif"><font style="box-sizing: inherit; outline: none; max-width: 100%; overflow: hidden; color: rgba(0, 0, 0, 0.87); font-size: 16px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;">∴</font></font>in each of three cases x² is either of the form 5q or 5q+1 or 5q+4 and for integer q.