Question

show that any odd integer is in the form of 4q+1,4q+3 where q is integer​

Answer 1

Answer:

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.

Step-by-step explanation:

Answer 2

If q is an integer

4q is divisible by 2

There for 4q is even number

Even +1 = odd

There for 4q +1 is odd number

Odd +2 = odd

There for 4q+1+2 = 4q+3 is odd number

Hope this will help....