Question

Q- Show that any positive even integer is of the form 8q, 8q + 2, 8q + 4, or 8q + 6, where q is some integer.

Answer 1

Let a be any positive integer and b = 8

By Euclid's division lemma

a = bq+r where 0≤ r<b

a = 8q +r
where r = 0,1,2,3,4,5,6,7

So a = 8q = even

a= 8q+1 = odd

a= 8q+2 = even

a = 8q+3= odd

a = 8q + 4= even

a = 8q + 5 = odd

a = 8q + 6 = even

a = 8q + 7 = odd

∴ 8q, 8q + 2, 8q + 4, or 8q + 6 are even

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

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



$\huge{ \green{thanks}}$