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

Best Answer

Let us start with taking a, where a is a positive odd integer. 

We apply the pision algorithm with a and b = 4.

Since 0  r < 4, the possible remainders are 0, 1, 2 and 3.

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

However, since a is odd, a cannot be 4q or 4q + 2 (since they are both pisible by 2).

Therefore, any odd integer is of the form 4q + 1 or 4q + 3.