Question

Show that any postive odd integer is of the form of 4q+1or4q+3,where q is some integer

Answer for my question please ​

Answer 1

Let a be any positive integer 

  a = bq +r 

0 ≤ r < b

possible remainders are 0, 1, 2 , 3 

 this shows that a can be in the form of

4q, 4q+1, 4q+2, 4q+3 q is quotient 

as a is odd a  can't be the form of 4q or

4q+2 as they are even

so a ill be in the form of 4q + 1 or 4q+ 3

$\huge\boxed{Hense\:Proved !!!}$

Answer 2

Given any integer n, apply the quotient-remainder theorem to n with d = 4. This implies that there exist an integer quotient q and a remainder r such that n = 4q + r and 0 r < 4. But the only nonnegative remainders r that are less than 4 are 0, 1, 2, and 3. Hence n = 4q or n = 4q + 1 or n = 4q + 2 or n = 4q + 3 for some integer q.