Question

prove that any odd prime number is either of the form 4a+1 or 4a-1 where 'a' is a positive integer​

Answer 1

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.

Answer 2

Given : odd prime number

To Find   : prove that any odd prime number is either of form 4a+1 or 4a-1 where 'a' is a positive integer​

Solution:

Without loosing generality any natural number can be written in form

4q , 4q+ 1 , 4q  + 2  , 4q  + 3

4q  = 2 ( 2q)  is even  number  

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

=>   4a  + 1   odd  number

4q + 2  = 2( 2q + 2)  even  number

4q  + 3    =   2(2q + 1)  + 1     odd  number

4q  + 3    =  4q  + 4  - 1  

= 4(q  + 1)  - 1

= 4a  - 1          odd number  

Hence   any odd prime number is either of form 4a+1 or 4a-1  

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