Question

show that every positive odd integers is of the form (4q+1) or (4q+3) for some integer q​

Answer 1

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.

Answer 2

Let a be any positive integer and b=4

By Euclid's division lemma

a= bq +r where 0≤ r<b

and q be any integer,q≥0

a= 4q+r

Possible remainder = 0, 1, 2 and 3

because 0≤r<4

That is ,a can be 4q, 4q+1,4q+2 and 4q+3 , where q is the quotient. However , since a is odd,a can not be 4q and 4q+2 (since they are both divisible by 2

4q=2(2q)

4q+2= 2(2q+1)

Therefore , any odd integer is of the form 4q+1 or 4q+3