Question

Show that any positive even integer is of the form 6m , 6m+2 or 6m+4 where m is some integer

Answer 1

Here is your answer...
By euclids dividion lemma
A= bq + r where 0=or<r <b
Let b=6
Then r= 0,1,2,3,4,5
When r=0 then a= 6q +0
= 6q
When r=1 then a=6q +1

When r=2 then a=6q +2
When r=3 then a=6q + 3
When r= 4 then a= 6q +4.
When r=5 then a=6q +5

Now 6q = 2(3q)
So 6q is a multiple of two. And it is even
So, 6q+1 is odd. (even no +1 is always odd)
Similarly 6q +2(it is equal to2(3q+1)) and 6q +4 (equal to 2(3q +2))are even. And 6q + 3 and 6q +5 are odd