Show that every positive odd integer is of the form (4q+1) or (4q+3) for some integer q.

Show that every positive odd integer is of the form (4q+1) or (4q+3) f

1.	Show that every positive odd integer is of the form (4q+1) or (4q+3) for some integer q.
Answer» Let a = 4q + r : 0\xa0{tex}\\leq r < 4{/tex}\xa0{tex}\\therefore a = 4 q = 2 ( 2 q ) \\text { an even integer }{/tex}{tex}a = 4 q + 1 = 2 ( 2 q ) + 1 \\text { an odd integer }{/tex}{tex}a = 4 q + 2 = 2 ( 2 q + 1 ) \\text { an even integer }{/tex}{tex}a = 4 q + 3 = 2 ( 2 q + 1 ) + 1 \\text { an odd integer }{/tex}{tex}\\therefore {/tex}\xa0Every positive odd integer is of the form\xa0{tex}( 4 q + 1 ) o r ( 4 q + 3 ) \\text { for some integer }{/tex}