Question

show that any positive odd integer is of the form 4q+1 or4q+3 where q is some integer

Answer 1

Let a be any positive integer, take b=4 -
Using Euclid's Division Lemma-
a = bq + r ( 0 < = b < r)
a= 4q + r
Possible forms -
(1) a = 4q
(2) a = 4q + 1 (odd)
(3) a = 4q + 2
(4) a = 4q + 3 (odd)

Hence, positive odd integer is of the form 4q+1 or 4q+3...

Answer 2

Given, 4q
b=4 (r=0,1,2,3,) (0<=rIf r=0
Then 4q+0---------eq 1
If r=1
Then 4q+1----------eq 2
If r=2
Then 4q+2----------eq 3
If r=3
Then 4q+3-----------eq 4


From the equations 2,4
We can show that any positive odd integer is of the form 4q+1 or 4q+3 where Q is some integer.