Question

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

Answer 1

Leta be any positive integer andb= 2

Then byEuclid 's division Lemma,there exist integersqandrsuch that

a= 2q+rwhere 0 ≤r

Now 0 ≤r r ≤ 1

⇒r= 0 orr= 1 [∵ris an integer]

∴a= 2qora= 2q+ 1

Ifa= 2q, thenais an even integer.

We know that an integer can be either even or odd.So, any odd integer is of the form 2q+ 1.

2) Letabe any positive integer andb= 4. Then by Euclids algorithm,

a= 4q+rfor some integerq≥ 0, andr= 0, 1, 2, 3So,a= 4qor 4q+ 1 or 4q+ 2 or 4q +3 because 0 ≤rNow, 4qi.e., 2(2q) is an even number∴4q+ 1 is an odd number.4q+ 2 i.e., 2(2q+ 1) which is also an even number.∴ (4q+ 2) + 1 = 4q+ 3 is an odd number.Thus, we can say that any odd integer can be written in the form 4q+ 1 or 4q+ 3 whereqis some integer

Answer 2

To Show :

Every positive odd integer is of the form 2qbabr 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