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

Let a be any odd integer and b = 4. So by applying Euclid’s Division Lemma,

So we have a = 4q + r

Where r is in the range of 0 to 4.

Case I:

For r = 0

a = 4q

Case II:

For r = 1

a = 4q + 1

Case III:

For r = 2

a = 4q + 2

a = 2 (2q + 1)

Case IV:

For r = 3

a = 4q + 3

Since 4q and 4q + 2 is multiple of 2. Thus they are even numbers.

Hence any positive odd integers is in the form of 4q + 1 and 4q + 3.

Hence Proved.