Question

show that any positive odd integer is of the form of 8q +1, 8q +3 or 8q +5 where q is integer

Answer 1

let us start with with taking a, where a is a positive odd integer. We apply the division algorithm with a and b = 8. since 0 ≤ r < 8 the possible remainders are 0,1,2,....7. That is a can be 8q or8q+1 or 8q+2 or 8q+3 or 8q+4or 8q+5 or 8q+6 where q is quotient. How ever since a is odd a cannt be 8q or 8q+2 or 8q+4 (since they are divisible by 2). There fore, any odd integer is of the form 8q+1, 8q+3, 8q+5 or 8q+7.

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

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

$\huge{ \green{thanks}}$