Question

show that any positive even integer is of the form 4q or 4q+2 where q is some integer

Answer 1

let the reminder be 0,1,2,3
r=0
4q

r=1
(4q+1)2
16q+1-8q
4q(4-2)+2
4q+2

r=2
(4q+2)2
16q+4-16
4q(4-4)+4
4q+4

r=3
(4q+3)2
16q+9-24
4q(4-6)+9

Answer 2

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

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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