If p is an even integer and q is an odd integer, which of the following must be an odd integer?
Let's analyze the options:
p/q: The result of dividing an even integer (p) by an odd integer (q) will always be an even integer divided by an odd integer, which results in an even integer. So, p/q is even.
pq: The product of an even integer (p) and an odd integer (q) will always be even since even * odd = even. So, pq is even.
2p + q: This expression involves adding an even integer (2p) and an odd integer (q). When you add an even integer and an odd integer, the result is always an odd integer. So, 2p + q is odd.
2(p + q): This expression involves adding an even integer (2) to the sum of an even integer (p) and an odd integer (q). The sum of p and q could be either even or odd, depending on the specific values of p and q. However, multiplying an even integer (2) by either an even or an odd integer will result in an even integer. So, 2(p + q) is even.
Therefore, the only expression that must be an odd integer is 2p + q.
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