Question

Show that any positive odd integer is of the form 8q+1 or 8q+3 or 8q+5 or 8q+7 for some integer q.

Answer 1

Acc. to lema
a=bq+r
o=<r<b
b=8
o=<r<8
r=0,1,2,3,4,5,6,7

if,,r=0
a=8q+o
=8q
if..r=1
a=8q+1
follow as per this by taking r=2,3,4,5,6,7 and then pick out odd ones

Answer 2

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

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

let a be any positive integer

then

b=8

0≤r<b

0≤r<8

r=0,1,2,3,4,5,6,7

case 1.

r=0

a=bq+r

8q+0

8q

case 2.
r=1
a=bq+r

8q+1

case3.

r=2

a=bq+r

8q+2

case 4.

r=3

a=bq+r

8q+3

case 5.

r=4

a=bq+r

8q+4

case 6.

r=5

a=bq+r

8q+5

case7.

r=6

a=bq+r

8q+6

case 8

r=7

a=bq+r

8q+7

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

$\huge{ \green{thanks}}$