Question

show that any positive or integer in the form 4q+1 or 4q+3

Answer 1

HEY

HERE IS ANSWER.

Question -

show that any positive or integer in the form 4q+1 or 4q+3

Now

Solution -

let a be any positive integer.

hence

b=4.

we know a formula


a=bq+r

and we are also known to this formula

0≤r<b

hence

0≤r<4


hence

r=0,1,2,3.

now

case 1
______

r=0

a=bq+r

a=4q+0

a=4q

case 2.
______

a=bq+r

r=1.

a=4q+1

case 3
______

a=bq+r
r=2

a=4q+2


case 4
______

a=bq+r

r=3

a=4q+3

hence from above it is proved
that any positive or integer in the form 4q+1 or 4q+3.


HOPE IT HELPS

THANKS

Answer 2

let a be any positive odd integer and b =4
Then, using Euclid ' d div. lemma that exist integers q and r such that
a= 4q+r where 0<= r <4
therefore r can be 0,1,2,& 3
* a=4q
* a=4q+1
* a = 4q+2
*a=4q+3

we should neglect the 1st & 3rd result coz, we get only an even number when we multiply an odd by an even number
hence , any positive odd integer can be expressed in the form of 4q+1 & 4q +3