Question

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

Answer 1

Let a be a positive odd integer 

Using the division algorithm on a and b = 4

a = 4q + r Since 0 ≤ r < 4, the possible remainders are 0,1, 2 and 3

∴ a can be 4q or 4q + 1 or 4q + 2 or 4q +3 , where q is the quotient 

Since a is odd ,a cannot be 4q + 4q + 2

∴ Any odd integer is of the form 4q + 1 or 4q + 3 , where q is some integer.