Question

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

Answer 1

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we have to prove that every positive even integer is of the form 2q and every odd positive integer is of the form 2q + 1.

we know that,

a= bq + r
where a is some integer and b is divisor.
q is quotient and r is remainder.

now, let the positive integer ( even ) be a.
b would be equal to 2.

the remainder shall be less than 2.

a= bq + 0
a= 2q --------> which is even. any number multiplied by 2 will become even. ( 1st part proved )

but if 1 is added it would give us an odd number.

so,

adding 1 we get
a= bq + 1
which is an odd positive integer. ( 2nd part proved )

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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