Question

Show that every positive odd integer is of the form (4q + 1) or (4q + 3), where q is some integer.

Answer 1

According to Euclid's division lemma, 
a = bq + r  where  0 ≤ r ≤ b
Now, let a be an odd positive integer and b = 4.
When 0 ≤ r ≤ 4 so, the possible values of r will be 0, 1, 2, 3.
Now, the possible values of a will be thus, 4q, 4q + 1, 4q+2, 4q +3 where q is an integer.
But, we already know that a is an odd positive integer.
So, a will be 4q + 1 and 4q + 3.