Question

Show that every positive odd integer is the former, (4q+1)or (4q+3) for some integer q

Answer 1

Let N be a positive Odd integer.
So N-1 is an even integer.  So N-1 = 2 p  for some integer p.

So N = 2 p + 1

Now p can be an even integer or an odd integer.
     Let p be an odd integer.  Then  p = 2 q + 1 for some integer q.
           So  N = 2 (2 q +1) + 1 = 4 q + 3

     Let  p be an even integer.  Then p = 2 q for some integer q.
            So  N = 2 ( 2 q) + 1 = 4 q + 1

Hence every positive odd integer is in the form of either 4q+1 or 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

THANKS

#BeBrainly.