Question

If a prime number, \(p,\) is squared and the result is added to the next prime number greater than \(p\) which of the following integers could be the resulting sum? Indicate all such integers. A. $$3$$ B. $$4$$ C. $$7$$ D. $$14$$ E. V$$58$$ F. $$60$$ G. $$65$$ H. $$69$$

Short answer

The correct answers are C. \(7\) and D. \(14\)

Step-by-step solution

Identify the first few prime numbers and find the next prime number of each

The first few prime numbers are \(2, 3, 5, 7, 11, 13, 17, 19, 23, \ldots\). The next prime number greater than each of these are \(3, 5, 7, 11, 13, 17, 19, 23, 29, \ldots\) respectively.

Square each prime number and add it to the next prime number

Calculating the square of each prime number and adding it to the next prime number, we get the following: \(2^2+3=7\), \(3^2+5=14\), \(5^2+7=32\), \(7^2+11= 62\), \(11^2+13=134\) and so on.

Compare the result with the given options

For each of these sums, check if they are in the given options. If they are, mark that option as a valid answer. From the results, it can be seen that 7 (from \(2^2+3=7\)) and 14 (from \(3^2+5=14\)) are in the given options.

Key concepts

Square of a Number

The square of a number is calculated by multiplying the number by itself. For instance, the square of 2 is computed as: \[ 2^2 = 2 \times 2 = 4 \] This operation is fundamental in mathematics, and it's used in many areas, such as geometry and algebra.
When squaring a prime number, it highlights how multiplication works by showing the repeated summation of a number. Furthermore, squaring forms the basis for many advanced problem-solving techniques in mathematics.

• To square a number means to find its area when that number represents the side length of a square.
• Squaring is used in various formulas, like the Pythagorean theorem in geometry.

This operation is critical not only to understanding simple multiplications but also to comprehending more complex mathematical concepts.

Integer Sums

Integer sums involve adding whole numbers, which are also known as integers. These include all positive and negative numbers, including zero.
Understanding how to properly add integers is crucial for solving many algebraic equations and understanding number patterns.
In the context of the exercise, the sum is derived by adding the square of a prime number with the next immediate prime number.
For example:

• First, find the square of a prime number, like 3: \( 3^2 = 9 \)
• Next, add it to the next prime: \( 9 + 5 = 14 \)

This procedure helps to determine which integers could be potential outcome sums in the given problem set. Practicing integer sums can help develop basic math skills crucial for solving more sophisticated problems.

Mathematics Problem Solving

Mathematics problem-solving involves applying logical steps and mathematical concepts to reach a solution.
The key is to break down the problem into smaller, manageable tasks.
Using strategies such as understanding the problem, planning how to solve it, carrying out the plan, and reviewing the solution can be highly effective. In this exercise, problem-solving skills are used to:

• Identify the prime numbers
• Execute the square of a prime number
• Add the next consecutive prime number

Breaking down the steps, as shown in the example, demonstrates a clear path to the solution. This structured approach is invaluable in tackling not just math problems but problems in various fields.

Prime Number Sequence

Prime numbers are numbers greater than 1 with no divisors other than 1 and itself. Identifying prime numbers can be a critical skill in mathematics as they are the building blocks of number theory.
The sequence of prime numbers starts as follows: 2, 3, 5, 7, 11, 13, without any obvious pattern, which makes their study intriguing.
Starting from a given prime, finding the next prime is a crucial step in many mathematical problems, including this exercise. This sequence naturally forms the basis for understanding more complex ideas in math, like cryptography and prime factorization.
By understanding how to progress through the prime number sequence, students can identify and leverage these properties in broader problem-solving contexts.