Question

show that any positive even integer can be written in the form 6q, 6q +2,or 6q+4 where q is an even integer

Answer 1

Let a be any positive even integer.

let b = 6( divisor )

Therefore by Euclid's Division Algorithm

a = 6q + r where 0<=r<b
but a is even
therefore r = 0,2,4

Case 1 r = 0

a = 6q

Case 2 r = 2

a = 6q + 2

Case 3 r = 4

a = 6q + 4

Form the above Cases
We can say

Hence Proved

Answer 2

Let a be any positive integer

By division algorithm

a = 6q + r, where 0 ≤ r < 6

a = 6q, 6q + 1, 6q + 2, 6q + 3, 6q + 4, 6q + 5

But a is an even integer

a = 6q, 6q + 2, 6q+ 4