Question

Write whether every positive integer can be of the form $4q + 2$, where $q$ is an integer.
A. Yes
B. No
C. Ambiguous
D. Data Insufficient

Answer 1

Hint:
We will begin by taking 2 common from the given expression, $4q + 2$, which will give us $2\left( {2q + 1} \right)$. Now, this will imply that only even numbers can be expressed as $4q + 2$, where $q$ is an integer. Then, we will take any odd number to prove our statement that no odd number can be expressed as $4q + 2$.

Complete step by step solution:
We have to check if every positive number can be expressed as $4q + 2$, where $q$ is an integer.
The expression, $4q + 2$ can be written as $2\left( {2q + 1} \right)$
Hence, the number of the form $4q + 2$ will always be an even number because it has a multiple of 2.
So, odd numbers cannot be represented in the form $4q + 2$
For example, consider the number 9 which cannot be expressed as of the form $4q + 2$
Let if 9 can be expressed as $4q + 2$
That is $9 = 4q + 2$
Which implies
$
  4q = 7 \\
  q = \dfrac{7}{4} \\
$
But, $q$ has to be an integer which is a contraction to the value we have got in the previous step as $\dfrac{7}{4}$ is not an integer.
Therefore, we can say that not every positive integer can be of the form $4q + 2$, where $q$ is an integer.

Hence, option B is correct.

Note:
If $a$ be a positive number, then if we divide it by 4, then by Euclid’s algorithm, we have $q$ as quotient and $r$ as remainder such that $a = 4q + r$, where $0 \leqslant r < 4$. Hence, every positive integer can be expressed as $a = 4q + r$, such that $r = 0,1,2,3$