Question

If $3p+2q=r,$ where $p,q,r$ are prime numbers, then which of the following statements is (are) possible

• A. $p$ can be an odd number
• B. $p$ can be an even number
• C. $q$ can be an odd number
• D. $q$ can be an even number

Answer 1

Answer: (A), (C), (D)

The correct options are
A $p$ can be an odd number
C $q$ can be an odd number
D $q$ can be an even number

$3p+2q=r$
Here, $r$ is definitely an odd number.

So, there are two cases
Case $:1$
$3p$ is odd and $2q$ is even
Now, here $p$ will be odd and $q$ can be odd or even.
Case $:2$
$3p$ is even and $2q$ is odd
Since, $2q$ can not be odd.
So, case $2$ is not possible.

Therefore, $p$ will be an odd number and $q$ can be an odd or even number.