Question

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

Answer 1

Dear Student,

Let a be any positive integer and b = 4. Then by Euclid’s algorithm,

a = 4q + r for some integer q ≥ 0, and r = 0, 1, 2, 3

So, a = 4q or 4q + 1 or 4q + 2 or 4q + 3 because 0 ≤ r < 4

Now, 4q i.e., 2(2q) is an even number

∴4q + 1 is an odd number.

4q + 2 i.e., 2(2q + 1) which is also an even number.

∴ (4q + 2) + 1 = 4q + 3 is an odd number.

Thus, we can say that any even integer can be written in the form of 4q or 4q + 2 where q is a whole number.

Regards