Given p is an even integer and q is an odd integer, let's examine each of the listed possibilities to determine which one must be an odd integer.
A. p/q: This expression does not necessarily result in an integer. Since p is an even number and q is an odd number, the division of p by q may result in a rational number.
B. pq: Multiplying an even integer (p) by any integer (including an odd integer q) will result in an even integer. This comes from the property of even numbers, that they can be expressed as 2n, where n is an integer. Consequently, multiplying the even number by any other number retains its evenness.
C. 2p+q: This expression will always result in an odd integer. The reason for this comes from the properties of even and odd numbers. An even number (2p) added to an odd number (q) will always result in an odd number.
D. 2(p+q): Here the expression within the bracket is the sum of an odd and even number, which results in an odd integer. When this odd number is multiplied by 2, the result would always be even.
E. 3p/q: Much like option 'A', this would involve dividing an even number (3p) by an odd number (q). This does not necessarily yield an integer value.
From the above, we can conclude that Option C (2p+q) is the expression that must always yield an odd integer under the given conditions.
Answer: C. 2p+q