Question

Show that every positive even 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

heya yor anser is _______________
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Let a be any positive integer and b=2. Then, by Euclid's division lemma there exist integers q and r such that

 a = 2q + r , where 0 </=  r < 2

now, 0 </= r <2

=> 0</= r</= 1

=> r=0 or, r=1 ( because r is an integer)

therefore a =2 q, then a is an even integer.

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

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