Question

The average of 3 prime numbers lying between 47 and 74 is \(191 / 3 .\) The greatest possible difference between any two out of the 3 prime numbers is : (a) 12 (b) 14 (c) 18 (d) can't be determined

Short answer

Based on the step-by-step solution, the greatest possible difference between any two out of the three prime numbers with an average of 191/3 is 18.

Step-by-step solution

Find all prime numbers between 47 and 74

The prime numbers between 47 and 74 are: {53, 59, 61, 67, 71}. Make a note of these prime numbers and proceed to the next step.

Find three prime numbers with an average of 191/3

In this step, you will find the three prime numbers whose average is 191/3 (or 63.666). Since the average of three numbers is a sum of the numbers divided by 3, we can write the equation: \([P_1 + P_2 + P_3] / 3 = 191/3\) Thus, we are looking for the triplets of prime numbers (P1, P2, P3) whose sum is 191. We can check different combinations of the given prime numbers from the previous step to see if their sum is 191: 53 + 59 + 61 = 173 53 + 59 + 67 = 179 53 + 59 + 71 = 183 53 + 61 + 67 = 181 53 + 61 + 71 = 185 53 + 67 + 71 = 191 Now that we have found the three prime numbers (53, 67, 71) whose sum is 191, we can proceed to find the greatest possible difference.

Calculate the greatest possible difference

With the three prime numbers 53, 67, and 71, we can calculate the greatest possible difference between any two of them as follows: 71 - 53 = 18 71 - 67 = 4 67 - 53 = 14 The greatest possible difference is 18. So the correct answer is (c) 18.

Key concepts

Quantitative Aptitude

Quantitative aptitude is a measure of an individual's ability to perform numerical calculations with speed and accuracy. Understanding quantitative aptitude is essential for solving a wide range of mathematical problems, from simple arithmetic to more complex equations involving variables.

In educational contexts, such as competitive exams or standardized tests, quantitative aptitude encompasses various mathematical domains including algebra, geometry, number systems, and arithmetic. Students are often required to demonstrate proficiency in these areas under time constraints, which adds an additional layer of challenge to the exercises. Problems like finding the average of prime numbers not only require knowledge of prime number calculation and average computation but also test one’s ability to apply multiple mathematical concepts simultaneously to identify the correct solution.

Prime Number Calculation

The calculation of prime numbers is fundamental in various fields of mathematics and computer science. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Understanding how to find and work with prime numbers is a key skill for tackling a myriad of mathematical challenges.

For instance, when solving the example problem, the first step is to identify all prime numbers within a specific range. This can be achieved using methods such as trial division or employing prime number theories that help predict the likelihood of primality. Recognizing prime numbers and their properties can lead to more efficient problem-solving strategies, as it narrows down the potential candidates that can be used to fulfill given conditions, such as reaching a specific average.

Average Calculation

The concept of average, also known as the arithmetic mean, is a key component in statistical and mathematical analysis. Calculating the average involves adding a group of numbers together and then dividing by the count of those numbers. This calculation is crucial for determining central tendencies or representing a set of data with a single value that summarizes its overall character.

In the context of the exercise, understanding the relationship between the sum of three prime numbers and their average allows us to form an equation and solve for the desired values. This equation represents a balance point and is instrumental in average problems that may appear in homework exercises and real-world applications alike. The ability to compute averages quickly and accurately is therefore a valuable skill in quantitative aptitude.