Question = Is pq + 1 = odd number?

Statement 1 = p - q = 5
p = 5 + q
From question, pq + 1 = odd number.
(5 + q ) * q + 1 = ?
$$5q\ +\ q^2\ +\ 1=?$$
There is no specific information about the value of q, hence statement 1 is INSUFFICIENT.

Statement 2 = p = 7
From question pq + 1 = odd number
7q + 1 = ?
There is no specific information about the value of q, hence statement 2 is INSUFFICIENT.
COMBINING STATEMENT 1 AND 2 TOGETHER.
$$\left(5q\ +\ q^2\ +\ 1\right)+\left(7q\ +\ 1\right)=0$$
$$12q\ +\ q^2\ +\ 2=0$$
$$\ q^2\ +\ 12q\ +\ 2=0$$
Using the quadratic information, $$x\ =\frac{\left(-b\ \frac{+}{-}\sqrt{b^2-4ac}\right)}{2a}$$
Where a = 1, b = 12, c = 2 and x = q
$$q\ =\frac{\left(-12\ \frac{+}{-}\sqrt{12^2-\left(4\ \cdot\ 1\ \cdot\ 2\right)}\right)}{2\ \cdot\ 1}$$
$$q\ =\frac{\left(-12\ \frac{+}{-}\sqrt{144\ -\ 8}\right)}{2\ }$$
$$q\ =\frac{\left(-12\ \frac{+}{-}\sqrt{144\ -\sqrt{8}\ }\right)}{2\ }$$
$$q\ =\frac{\left(-12\ \frac{+}{-}12-\sqrt{8}\ \right)}{2\ }$$
$$q\ =\frac{\left(-12\ \frac{+}{-}12-\sqrt{8}\ \right)}{2\ }\ or\ q\ =\ \frac{\left(-12\ -12\ -\sqrt{8}\right)}{2}$$
$$q\ =\frac{-\sqrt{8}\ }{2\ }\ or\ q\ =\ \frac{\left(-24\ -\sqrt{8}\right)}{2}$$
$$q\ =\frac{-\sqrt{8}\ }{2\ }\ or\ q\ =\ -12\ -\sqrt{8}$$
After combining statement 1 and 2 together, the value of q is not yet specific, hence the two statement together are not SUFFICIENT

Option E is CORRECT.

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Modifying the question:
pq + 1 is odd only when pq is even. So, the question is asking whether either p or q is an even prime number. Since the only even prime number is 2, the question is asking whether p or q is equal to 2.


Condition 1):
For p - q = 5, either p or q must be even. Since the only even prime number is 2, we must have p = 7 and q = 2.
Thus, condition 1) is sufficient.

Condition 2)
Since it provides no information about q, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A