Question

show that any positive odd integer is of the form 4q+1 or 4q+3, where q is some integer

Answer 1

let a be any positive integer..
b=4q+r where r and q are some unique integer..
r=0,1,2and3


b=4q,4q+1,4q+2 and 4q+3.....

but as a is a odd integer...

b=4q+1 & 4q+3




hope this will help you out

Answer 2

case -1
let 4q+1 be x
and let q=m
4m+1=x
squaring on both sides
(4m+1)^2 = x^2
(4m)^2 +(1)^2 +2(4m)(1)= x^2
16m^2 + 1 +8m= x^2
4(8m^2+2m)+1 =x^2.
4(q)+1= x^2. (because q=8m^2+2m)

same way you can do 4q+3