Question

Which of the following CANNOT be the sum of two integers that have a product of 30?

• A. 31
• B. 17
• C. -11
• D. -13
• E. -21

Hint:

For "CANNOT be" questions, the process is one of elimination. Systematically list all possibilities and check each option against your list. Don't forget to include negative integer pairs when finding factors.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
We are looking for a number in the options that cannot be formed by adding two integers, let's call them \(m\) and \(n\), where their product \(m \times n = 30\). We need to find all possible integer pairs whose product is 30 and then calculate their sums.
Step 2: Detailed Explanation:
1. List all integer pairs with a product of 30.
We need to consider both positive and negative pairs.
\begin{itemize} \item Positive Pairs: \begin{itemize} \item 1 and 30 \item 2 and 15 \item 3 and 10 \item 5 and 6 \end{itemize} \item Negative Pairs: \begin{itemize} \item -1 and -30 \item -2 and -15 \item -3 and -10 \item -5 and -6 \end{itemize} \end{itemize} 2. Calculate the sum for each pair.
\begin{itemize} \item \(1 + 30 = 31\) \item \(2 + 15 = 17\) \item \(3 + 10 = 13\) \item \(5 + 6 = 11\) \item \(-1 + (-30) = -31\) \item \(-2 + (-15) = -17\) \item \(-3 + (-10) = -13\) \item \(-5 + (-6) = -11\) \end{itemize} 3. Check the options against the possible sums.
\begin{itemize} \item (A) 31: This is a possible sum (from 1 + 30). \item (B) 17: This is a possible sum (from 2 + 15). \item (C) -11: This is a possible sum (from -5 + -6). \item (D) -13: This is a possible sum (from -3 + -10). \item (E) -21: This sum is not in our list of possibilities. \end{itemize} Step 3: Final Answer:
The value -21 cannot be the sum of two integers whose product is 30.