Question

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

Answer 1

let a= 4q+r ,where q and r are some integars ( as Euclid's algorithm )
now , r= 1,2,3
but for positive odd intereg ,r=1,3 only
:. when r=1 ,
a=4q+1
and, when r=3
then, a=4q+3

Answer 2

Step-by-step explanation:

Let a be the positive integer.

And, b = 4 .

Then by Euclid's division lemma,

We can write a = 4q + r ,for some integer q and 0 ≤ r < 4 .

°•° Then, possible values of r is 0, 1, 2, 3 .

Taking r = 0 .

a = 4q .

Taking r = 1 .

a = 4q + 1 .

Taking r = 2

a = 4q + 2 .

Taking r = 3 .

a = 4q + 3 .

But a is an odd positive integer, so a can't be 4q , or 4q + 2 [ As these are even ] .

•°• a can be of the form 4q + 1 or 4q + 3 for some integer q .

Hence , it is solved .