Question

Show that any odd positive integers is in the form of 4k+1 or 4k+3,k ez.​

Answer 1

Answer:

..........ok............@ .....

Answer 2

Answer:

Let n be any odd prime. If we divide any n by 4, we get

n=4k+r

where 0≤r≤4 i.e., r=0,1,2,3

∴eithern=4korn=4k+1

or n=4k+2orn=4k+3

Clearly, 4n is never prime and  

4n+2=2(2n+1) cannot be prime unless n=0

(since, 4 and 2 cannot be factors of an odd prime).

∴ An odd prime n is either of the form  

4k+1or4k+3

But 4k+3=4(k+1)−4+3=4k  

′

−1

(where  k  

′

=k+1)

∴ An odd prime n is either of the form  

4k+1or(4k+3)i.e.,4k  

′

−1