Question

Show that any positive odd integer is of the form 4q+1 or 4q+3 where q is a positive integer??

Pleaase answer fasti have exam tommorow!!

Answer 1

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let a be any positive integer

then

b= 4

a= bq+r

0≤r<b

0≤r<4

r= 0,1,2,3

case 1.

r=0

a= bq+r

4q+0

4q

case 2.

r=1

a= 4q+1

6q+1

case3.

r=2

a=4q+2

case 4.

r=3

a=4q+3

hence from above it is proved that any positive integer is of the form 4q, 4q+1,4q+2,4q+3

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Answer 2

solutions :-

Let m be any positive integer

then

b= 4

m= bq+r

0≤r<b

0≤r<4

r= 0,1,2,3

case 1.

r=0

m= bq+r

4q+0

4q

case 2.

r=1

m= 4q+1

6q+1

case3.

r=2

m=4q+2

case 4.

r=3

m=4q+3

Hence

it is proved that any positive integer is of the form 4q, 4q+1,4q+2,4q+3

Thanks

@Anushka