Question

Show that any positive odd integer
Is of the form 4q +1 or 4q+3

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.

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Answer 2

Explanation:

let us start with taking s where a is positive odd integers we apply the division algorithm with a and b = 4

since 0 <_ r < the possible remainder are 0 ,1,2 and 3

this can be 4q or 4q +1 or 4q +2 or 4q +3

where q is the quotient how ever , since odd a cannot be 4q or 4q + 2

any odd integers is form of 4q + 1 or 4 q + 3