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

Solution:
No, every positive integer cannot be only of the form 4q + 2.
Justification:
Let a be any positive integer. Then by Euclid’s division lemma, we have
a = bq + r, where 0 ≤ r < b

Putting b = 4, we get
a = 4q + r, where 0 ≤ r < 4
Hence, a positive integer can be of the form,
4q, 4q + 1, 4q + 2 and 4q + 3.​

No, every positive integer cannot be of the form 4q + 2, where q is an integer.

Justification:

All the numbers of the form 4q + 2, where ‘q’ is an integer, are even numbers which are not divisible by ‘4’.

For example,

When q=1,

4q+2 = 4(1) + 2= 6.

When q=2,

4q+2 = 4(2) + 2= 10

When q=0,

4q+2 = 4(0) + 2= 2 and so on.

So, any number which is of the form 4q+2 will give only even numbers which are not multiples of 4.

Hence, every positive integer cannot be written in the form 4q+2