Question

5# Show that every positive odd integer is of the form 4q +1 and 4q+3,where q is some integer​

Answer 1

Step-by-step explanation:

Given :-

A positive integer

To find :-

Show that "Every positive odd integer is of the form 4q +1 and 4q+3,where q is some integer".

Solution :-

We know that

Euclid's Division Lemma:

For any two positive integers a and b there exist two positive integers q and r satisfying a = bq+r, 0≤r<b.

Let consider a = 4q+r ------------(1)

The possible values of r = 0,1,2,3

I) If r = 0 then

a = 4q+0

=>a = 4q

=> a = 2(2q)

=> a = 2m -----------(2)

Where m = 2q

ii) If r = 1 then

a = 4q+1-----------(3)

iii) If r = 2 then

a = 4q+2

=> a = 2(2q+1)

=> a = 2m ----------(4)

Where m = 2q+1

iv) If r = 3 then

a = 4q+3 -----------(5)

From (2)&(4)

a is the positive even number.

From (3)&(5)

a is the positive odd number.

Every positive odd integer is of the form 4q +1 and 4q+3,where q is some integer.

Hence, Proved.

Used formulae:-

Euclid's Division Lemma:-

For any two positive integers a and b there exist two positive integers q and r satisfying a = bq+r, 0≤r<b.

Answer 2

Step-by-step explanation:

We have

Any positive integer is of the form 4q+1or4q+3

As per Euclid’s Division lemma.

If a and b are two positive integers, then,

a=bq+r

Where 0≤r<b.

Let positive integers be a.and b=4

Hence,a=bq+r

Where, (0≤r<4)

R is an integer greater than or equal to 0 and less than 4

Hence, r can be either 0,1,2and3

Now, If r=1

Then, our be equation is becomes

a=bq+r

a=4q+1

This will always be odd integer.

Now, If r=3

Then, our be equation is becomes

a=bq+r

a=4q+3

This will always be odd integer.

Hence proved.

I HOPE IT WILL HELPFUL TO YOU