Answer: D

We need to determine whether every number of $T$ is from $Y$, that is, whether every number of $T$ is even.

Statement I: The sum of any two numbers belonging to $T$ is even.

This means that all numbers in $T$ must have the same parity. They may all be even, or they may all be odd.

For example:

• If $T=\{2,4\}$, then the sum of any two numbers is even, and every number is from $Y$.
• If $T=\{1,3\}$, then the sum of any two numbers is also even, but every number is from $X$, not from $Y$.

So, Statement I alone is not sufficient.

Statement II: If both $p$ and $q$ are picked from $T$, then $(p-1)q$ is even.

If all numbers in $T$ are even, then $q$ is even, so $(p-1)q$ is even.

If all numbers in $T$ are odd, then $p-1$ is even, so $(p-1)q$ is also even.

Thus, Statement II is satisfied when all numbers are even and also when all numbers are odd. Hence, Statement II alone is not sufficient.

Using both statements together:

Even together, the statements allow both possibilities:

• $T$ may contain only even numbers, in which case every number of $T$ is from $Y$.
• $T$ may contain only odd numbers, in which case no number of $T$ is from $Y$.

Therefore, the question cannot be answered even using both statements together.