Question

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

Answer 1

Answer:

Let 'a' be any positive integer,then

b= 6

a= bq+r  ;  0≤r<b

0≤r<6

∴ r= 0,1,2,3,4,5

when r=0,

a= bq+r

6q+0=6q →Even

when r=1,

a= 6q+1

6q+1 → Odd

when r=2,

a=6q+2 → Even

when r=3,

a=6q+3 → Odd

when r=4,

a=6q+4 → Even

when r=5,

a=6q+5 → Odd

Thus, a = 6q or, 6q +1 or, 6q + 2 or, 6q + 3 or, 6q + 4 or, 6q +5.

But here, 6q +1, 6q + 3, 6q +5 are the odd integers.

Therefore, 6q or, 6q + 2 or, 6q + 4 are the forms of any positive even integers.