**Question:**

Show that any positive odd integers is of the form 4*q*+1 or 4*q*+3 where q is a positive integer.

**Solution:**

Here we have to prove that for any positive integer *q*, the positive odd integer will be form of 4*q*+1 or 4*q*+3.

Now let us suppose that the positive odd integer is *a* then by Euclid’s division rule

*a* = 4*q* + *r* ……(1 )

Where *q* (quotient) and *r* (remainder) are positive integers, and

We are putting the values of *r* from 0 to 3 in equation (1), we get

But we can easily see that 4*q* and 4*q*+2 are both even numbers.

Therefore for any positive value *q*, the positive odd integer will be the form of 4*q*+1 and 4*q*+3.