Show that any positive odd integer of the form `(4m +1)` or `( 4m +3)` for some integer `m`.

Explanation

Let n be an arbitrary odd positive integer.
On dividing n 4 , let m be the quotient and r be the remainder. So ,by Eulid's division lemma, we have
n = 4m +r , where ` 0 ge lt 4`
n = 4m or ( 4m +1) or ( 4m +2) or ( 4m +3)
Clearly , 4m and ( 4m +2) are even and since n is odd, so ` n ne 4 m and n ne ( 4m +2)`
n = ( 4m+1) or ( 4m +3) , for some integer m.
Hence, any positive odd interger is of the form ( 4m +1) or f( 4m+1) for some integer m.