Show that any positive odd integers is of the form 4q+1 or 4q+3 where q is a positive integer.
Here we have to prove that for any positive integer q, the positive odd integer will be form of 4q+1 or 4q+3.
Now let us suppose that the positive odd integer is a then by Euclid’s division rule
a = 4q + 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 4q and 4q+2 are both even numbers.
Therefore for any positive value q, the positive odd integer will be the form of 4q+1 and 4q+3.