If $p$ is an odd integer and $q$ is an even integer, which one of the following is an odd integer?
(A) $2 p+3 q$
(B) $3 p+2 q$
(C) $4 p+q$
(D) $2(p+3 q)$
(E) $p q$
Solution 1

Since $p$ is odd and $q$ is even, then $3 p$ is odd times odd (so is odd) and $2 q$ is even times even (so is even).

Therefore, $3 p+2 q$ is odd plus even, which is odd.

(Since we have found one possibility that is odd, we do not need to look at the others. We could check, though, that each of the others is always even.)

## Solution 2

We check for a particular case of $p$ and $q$, since the problem implies that the result is the same no matter what odd and even integers $p$ and $q$ we choose.

We test the five choices with $p=1$ and $q=2$, which are odd and even, respectively.

In this case, $2 p+3 q=8$ and $3 p+2 q=7$ and $4 p+q=6$ and $2(p+3 q)=14$ and $p q=2$.

Thus, the only possibility that is odd is $3 p+2 q$.

ANSWER: (B)

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