Question

Show that every positive integer is of the form 2q , and that every positive odd integer is of the form 2q+1 , where q is some integer.

Answer 1

To Show :

Every positive odd integer is of the form 2q and that every positive odd integer is of the form 2q+1, where q € Z .

Solution :

Let a be any positive integer.

And let b = 2

So by Euclid's Division lemma there exist integers q and r such that ,

a = bq+r

a = 2q+r (b = 2)

And now ,

As we know that according to Euclid's Division Lemma :

0 ≤ r < b

Here ,

0 ≤ r < 2

Here the possible values of r are = 0,1

=> 0 ≤ r < 1

=> r = 0 or r = 1

a = 2q+0 = 2q or a = 2q+1

And if a = 2q , then a is an integer.

We know that an integer can be either odd or even.

So , therefore any odd integer is of the form 2q+1.

#Hence Proved