Question

1.Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer. 2.Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

Answer 1

1] a=2q
squaring on both sides
a2=[2q]2
a2=4q2
a2=4m [where m=q2]

2] a=2q +1
sq. on bs
a2=[2q]2+1
a2=4q2+1
a2=4m where m=q2+1

Answer 2

$\huge \bf{ \red{hey}}$

$\huge{ \mathfrak{here \: is \: answer}}$

let a be any positive integer

then

b=2

0≤r<b

0≤r<2

r=0,1

case 1.

r=0

a=bq+r

2q+0

2q

case 2.
r=1
a=bq+r

2q+1

from above it is proved.



now

b=4

r=0,1,2,3

a=bq+r

case 1.

r=0

4q+0

4q

case2.

r=1

4q+1

case 3.

r=2

4q+2

case 4

r=3

4q+3

$\huge \boxed{ \boxed{ \pink{hope \: it \: helps}}}$

$\huge{ \green{thanks}}$