Question

Which of the following forms can be used to represent any positive even integer?
(Note: q > 0)

• A. $6q$
• B. $6q+2$
• C. $6q+3$
• D. $6q+4$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $6q$
B $6q+2$
D $6q+4$

Let 'a' be any positive integer and b = 6.
Then by Euclid's algorithm,
a = 6q + r, for some integer q >= 0 and
0 <= r <= 5
Then a can be,
6q , 6q + 1, 6q + 2, 6q + 3,
6q + 4, 6q + 5.

Since 'a' is even, 'a' cannot be 6q + 1, 6q + 3, 6q + 5 (since these are not divisible by 2).
Therefore, any positive even integer is of the form 6q, 6q + 2 or 6q + 4.