Question

If the product of two distinct integers is \(91,\) then which of the following could be the sum of the two integers? Indicate all such sums. A. $$-92$$ B. $$-91$$ C. $$7$$ D. $$13$$ E. $$20$$

Short answer

The possible sums are -92 and 20.

Step-by-step solution

Listing the factors of 91

First identify all the factors of 91. Factors of 91 are pairs of numbers that you could multiply together to get 91. Since 91 equals 7 times 13, the pairs of integers that multiply to become 91 are: (1,91), (-1,-91),(7,13), (-7,-13).

Compute the sums

Now, add each pair of factors to compute their sums. The sum of (1,91) is 92; the sum of (-1,-91) is -92; the sum of (7,13) is 20; the sum of (-7,-13) is -20.

Match the sums

From Step 2, we found four sums: 92, -92, 20, -20. Now compare these to the given options (A. -92, B. -91, C. 7, D. 13, E. 20). From this comparison, we find that the sums that match the options are -92 (option A) and 20 (option E). These are the possible sums of two distinct integers that multiply to 91.

Key concepts

Integer Properties

Understanding integer properties is crucial when solving math problems, especially those involving multiplication and addition. Integers are whole numbers that can be positive, negative, or zero. A product of two integers is also an integer. To find integers whose product is 91, you need to determine the possible pairs of integers whose multiplication results in that product. Identifying factors or divisors is an essential skill here.

Being familiar with factorization is helpful, which involves breaking down a number into its component integers or factors. For 91, the factor pairs are (1, 91), (-1, -91), (7, 13), and (-7, -13). Knowing that integers can be negative expands the possibilities when analyzing all potential factor pairs. Recognizing negative factor pairs is as important as recognizing their positive counterparts, as they can impact the eventual sum of the integers.

Problem-Solving Steps

Approaching math problems with a structured method can simplify even the most complex tasks. Start by identifying what the problem is asking, then break it into smaller, manageable steps.

• Step 1: Identify factors - To solve the problem, you first need to list all factor pairs of the given product, in this case, 91.
• Step 2: Sum the factors - Calculate the sum of each factor pair.
• Step 3: Compare sums - Match these sums against the possible answer choices.

These steps help in systematically narrowing down the correct answers to a problem. For this exercise, finding all pairs that multiply to 91 led to sums of 92, -92, 20, and -20. By comparing these sums to the given options, you can determine the correct solutions.

Math Reasoning

Math reasoning involves logical and analytical thinking to solve problems. This exercise specifically requires understanding relationships between numbers through the concepts of multiplication and addition.

When faced with a problem involving the product and sum of integers, you should think about various factor combinations and how they relate to the question asked. Visualizing factor pairs mentally or writing them out helps clarify their relationships. Through reasoning, you focus on whether given conditions, like a product of two integers, can yield particular sums.

In this case, logical reasoning where you match the given sum questions with calculated sums directs you toward the correct options. Thus, option A (-92) and option E (20) accurately reflect the calculations from the determined integer pairs.

GRE Test Preparation

Preparing for the GRE test involves strengthening your understanding of various math concepts, including integer properties and basic arithmetic operations such as multiplication and addition. This problem, like many on the GRE, tests your ability to apply mathematical knowledge to find solutions efficiently.

To prepare, focus on practicing problems that require recognizing and applying integer properties.

• Practice factorization regularly to understand how to break down numbers.
• Develop a step-by-step problem-solving strategy to ensure you find all possibilities.
• Work on enhancing your math reasoning skills through logical puzzles and exercises.

By honing these skills, you ensure that you can address similar questions quickly and accurately, giving you the confidence and ability to succeed on the GRE.