Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer.

Question

Philoid · Accepted Answer

No

By Euclid’s Lemma,

b = a × q + r, 0 ≤ r < a [Using dividend = divisor × quotient + remainder]

Here, b is any positive integer.

According to the question, a = 4

⇒ b = 4q + r where 0 ≤ r < 4

⇒ r = 0, 1, 2, 3

So, this must be in the form 4q, 4q + 1, 4q + 2 or 4q + 3.

Hence, every positive integer cannot be of the form 4q + 2.